Inquiry and problem solving
The last post about problem solving featured a discussion with Dan Meyer (@ddmeyer). It centred on the notion of openness in maths classrooms with Dan arguing that openness is "a spectrum, not a switch." I was not convinced by this argument in relation to problem solving.
Inquiry can be more or less open (see levels of inquiry), but problem solving only has the 'open middle'. While the the teacher poses the problem and knows the answer, the solving process can be carried out in different ways. Skills required by students relate to extracting relevant information from the problem, identifying similar problems that they have solved before, selecting methods, checking progress and verifying the solution. Inquiry has the potential for an open beginning, middle and end. The teacher will have an idea of different pathways that could arise, but they will not be exhaustive. Moreover, students' initial questions and findings might be novel. Key processes include questioning, noticing, conjecturing and proving.
The distinction is reinforced in the National Curriculum in England. Reasoning through inquiry and problem solving make up two of the three separate aims of the curriculum:
Unsurprisingly, the GCSE assessment objectives (AO), which are based on the National Curriculum, feature the same separation. AO2 covers reasoning, interpreting and communicating (which might be broadly classed as elements of inquiry) and AO3 lists problem solving steps (below).
The distinction came up again in a discussion I was having with a new head of a maths department (
@PythagnPi). She wrote to Inquiry Maths about how to develop mathematical reasoning: "I was thinking an inquiry approach would lead to encouraging the students to question what they do and why they do it, then lead to helping them with prompts for solving a problem. Do students not need to be on the path of fluency of inquiry before they can embark on the problem solving approach?" The question implies that mathematical inquiry is a precursor to problem solving.
During the discussion, one teacher said “it all depends on how you define a problem.” If you define a problem in terms of the questions that appear in public examinations, then, as Mike Ollerton (@MichaelOllerton) contends, problems are “pseudo-problems which undermine mathematical thinking and all that is creative in maths." They have one closed answer that students are required to find. Yet, as another participant in the discussion remarked, a problem set in a classroom does not necessarily imply the answer is known. In problem-based learning, for example, problems are open-ended, even if the starting point (the problem) is closed from the students' perspective. Geoff Wake (@geoffwake1), Associate Professor in Mathematics Education at the University of Nottingham, posted the following comment: "It's useful to think about the difference between solving a problem and problem solving." I took this to mean that ‘solving a problem’ is a restricted process with a closed beginning and end, but ‘problem solving’ is a creative process of applying generic skills, such as Polya’s heuristics, to an open-ended problem.
The discussion echoes a distinction between two different types of problems that Polya himself identifies in How to Solve It. On the one hand, a problem to find aims to "to find a certain object” and, in order to achieve the aim, the solver must know the problem’s principal parts, the unknown, the data and the condition. On the other hand, a problem to prove aims to show conclusively that a certain clearly stated assertion is true or false. Its principal parts are the hypothesis and the conclusion of the theorem to be proved or disproved. We note that, while the problem to find could be “theoretical or practical, abstract or concrete, serious problems or mere puzzles,” the problem to prove lies exclusively in the domain of mathematics.
It is a short step to linking a problem to find with the problem solving strand of the National Curriculum (including 'non-mathematical' contexts) and the problem to prove to the mathematical reasoning strand. However, the problem to prove is not synonymous with inquiry. In promoting students’ questions, exploration and conjectures, inquiry involves far more than a deductive proof of a theorem. While in Inquiry Maths lessons the teacher might help students formalise their ideas into a problem to prove, the starting point of inquiry (the prompt) cannot be likened to either of Polya’s problem types.
Paradoxically, inquiry could involve what Polya describes as routine problems. These focus on the mechanical performance of operations and "can be solved either by substituting special data into a formerly solved general problem, or by following step by step, without any trace of originality, some well-worn conspicuous example." When students select a regulatory card to practise a procedure, the teacher might suggest answering routine problems, although restricting students exclusively to this type of problem is, according to Polya, “inexcusable" and, we might add, antithetical to the principles of inquiry.
After this discussion, we must amend our characterisation of the relationship between inquiry and problem solving. We maintain the distinction – and even the separation – between classrooms in which problems to find (with their ‘open middles’) predominate and classrooms that emphasise inquiry processes linking exploration and deduction. However, problem solving can develop into inquiry when, for example, students change the conditions in the problem and study the relationship between the new solution and the old one. Nevertheless, I cannot conceive of a situation when or a reason why an open inquiry would be restricted to a problem-solving process. While inquiry skills enable students to attempt to solve problems, the converse is not true. Heuristics employed in problems to find would not, on their own, enable students to generate mathematical inquiry.

Andrew Blair 
February 2017