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Inquiry and curriculum

Fiona Hickey (@thefionahickey), a year 1 PYP teacher from
Canberra, asked Inquiry Maths about curriculum content and design as she promotes inquiry in her school: 
How do you decide which curriculum outcomes you will cover in a unit? Do you have a curriculum plan? 
Lesley Cowey (@CoweyLesley), a UK secondary school maths teacher, asked how inquiry is incorporated into schemes of learning: What fraction of your lessons are inquiry lessons? How do you integrate inquiry with more traditional lessons? 
A scheme of learning based on Inquiry Maths prompts requires an extended time on each part of the curriculum. A theme or topic might last for six or seven weeks, during which time the class could carry out two inquiries (see the outline at the bottom of the page). If the questions and observations at the start of the inquiry are rich and encompass the full potential of the prompt, then the teacher would aim to base the inquiry on them for the full time with teacher and students structuring the lessons using the regulatory cards. This might mean students are engaged in independent inquiry, but it might also mean the teacher plans a structured session to explain a specific procedure.
In setting the prompt just above the understanding of the class, the teacher expects students to require new procedures and concepts to make progress in the inquiry. At such times, students find new knowledge both meaningful and valuable. For example, when a year 8 class saw the area prompt, students identified the need to know how to calculate the area of a circle. In consequence, the teacher planned a lesson on explaining the formula. Thus, it is not a question of integrating inquiry with traditional lessons, but of integrating a teacher's explanation into inquiry when it is appropriate. As the inquiry comes to an end, the teacher designs pathways linked to the prompt in order to fill any 'holes' in the coverage of the curriculum.
An inquiry-based scheme of learning
Billy Adamson and Tony Germany of Thurston Community College (Bury St Edmunds, UK) have devised a scheme of learning for year 7 based on inquiry. They have incorporated Inquiry Maths prompts into the scheme, along with tasks and assessments. The outline (below) shows the extended blocks of time given to different parts of the curriculum.

Billy Adamson is the Head of Mathematics at Thurston Community College. You can follow Billy on twitter@Billyads_47.

Inquiry learning and the mathematics curriculum  
A national or state curriculum can seem an obstacle to teachers wishing to introduce inquiry processes. It appears to close down a teacher’s options. In a 2010 survey in 12 EU countries, the most common reason maths teachers gave for not introducing inquiry was that "there is not enough
time in the curriculum." Professor Raffaella Borasi (right) in her book
Learning Mathematics through Inquiry (1992) suggests that a rigid curriculum is incompatible with inquiry: "No fixed and established curriculum, however well constructed, could really respond to the needs of an instructional approach that stresses students’ independent learning" (p. 202). Open-ended inquiry
, Borasi argues, requires extreme flexibility in terms of curriculum content and choices.
Flexibility was a feature of the curriculum at John Dewey’s Chicago Laboratory School, which evolved on an experimental basis in response to children’s changing interests. In Schools of Tomorrow (1915), John and Evelyn Dewey contrast their "organic curriculum" growing out of a child’s experience to a systematised and standardised curriculum that ignores the needs of individual children. However, they also praise the cross-curricular projects of Public School 45 in Indianapolis that infused the state curriculum with “intrinsic meaning and value” (p. 77).
Even though the timetable hampers cross-curricular projects in most secondary (and even primary) schools today, dividing the day into separate subjects does not preclude inquiry. Indeed, the discipline of mathematics has partly developed though inquiry into its own representations, tools, forms of reasoning and language. As Artigue and Baptist acknowledge here, mathematics “creates its own objects and reality, and the questions raised by these objects have always been an essential motor of its development” (p. 5). In the same way, Inquiry Maths uses prompts internal to mathematics to generate inquiry.
Moreover, Inquiry Maths prompts are also internal to the mathematics curriculum in that they are designed to lead into pathways that are linked to national curricula. While they ‘cover’ prescribed content objectives, the order of that coverage, the time spent on different topics and the connections made between aspects of the curriculum are co-constructed with students. The order, time and connections evolve from the initial phase of questioning and noticing that students undertake when they first see the prompt. Thus, Inquiry Maths addresses the curriculum, but in a flexible way that promotes students' curiosity and initiative.
Read about curricula that promote mathematical inquiry here.