Inquiry and curriculum

Inquiry learning and the mathematics curriculum

A national or state curriculum can seem to be an obstacle to teachers wishing to introduce inquiry learning. A curriculum 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 (pictured) 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”.

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).

Inquiry Maths and the curriculum

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 lines of inquiry 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 and responds to students' curiosity and initiative.

Curriculum, knowledge, and inquiry

The article discusses the place of knowledge and the curriculum in England's 2019 school's inspection framework. The inspectorate's preference for knowledge-rich over skills-based curricula seems to put inquiry on the back foot.

However, as Dewey stressed throughout his writings, inquiry is not an anti-knowledge model. Inquiry teachers should use the curriculum to design students' experiences. Indeed, as the article argues, inquiry learning integrates two types of knowledge - disciplinary (the content of the curriculum) and epistemic (the way mathematicians create and validate knowledge ).

Curricula that promote mathematical inquiry

International Baccalaureate

Inquiry-based learning is central to the International Baccalaureate's curriculum. All three of the IB's programmes (PYP, MYP, and Diploma) integrate inquiry processes into the teaching and assessment of syllalbus content. For example, the IB Middle Years Programme (for students aged 11-16) promotes "both inquiry and application, helping students to develop problem solving techniques that transcend the discipline and that are useful in the world beyond school."

The IB's curriculum aims to ensure students understand the principles and concepts of mathematics and its use in modelling real-life situations. As the curriculum writers explain in the subject brief, the MYP is "tailored to the needs of students, seeking to intrigue and motivate them to want to learn" by studying authentic examples that are useful and relevant to their lives. In this way, the IB seeks to encourage sustained inquiry in schools.

For a critique of the IB's cycle of mathematical inquiry, read the article.

Australia

In the Rationale for the Australian F-10 mathematics curriculum, the designers hope to encourage teachers "to help students become self-motivated, confident learners through inquiry and active participation in challenging and engaging experiences."

The Australian government launched the reSolve: Maths by Inquiry programme in November 2015 to develop classroom resources in support of the curriculum. The programme has developed teaching sequences of structured inquiries for Foundation to Year 10. (Read our response to the announcement of the project here.)

England

The second of three aims of the 2013 National Curriculum in England is to ensure that all students "reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language."

While the writers of the curriculum might have envisaged short discrete tasks in their use of the word 'enquiry', it is still the case that the mathematical processes in their list could form the basis of a consistent inquiry pedagogy (see The difference between 'inquiry' and 'enquiry').

Frequently asked questions

Question

Fiona Hickey, a year 1 PYP teacher from Canberra (Australia), 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?

Response

Schools and departments that use Inquiry Maths will follow a curriculum plan. To maximise the potential of a prompt, the topics in the curriculum are combined into a broad theme that lasts longer than, for example, a two-week unit. This allows time to inquire more deeply, make connections between concepts, and develop lines of inquiry in a way that responds to the students' contributions during the question, notice, and wonder phase of the inquiry.

For example, in year 7 (grade 6), the schemes of learning might include six or seven weeks on calculations. The block of learning would focus on the four operations in a number of contexts, such as, with integers, decimals, and fractions. The teacher could launch each week with a separate prompt or extend an inquiry over more than one week if students' questions took the inquiry in the direction mapped out in the curriculum plan.

As the time set aside for a theme comes to an end, the teacher designs lessons to fill any 'holes' in the coverage of the curriculum.

Question

Lesley Cowey, a UK secondary school teacher of mathematics, asked how inquiry can be incorporated into schemes of learning: What fraction of your lessons are inquiry lessons? How do you integrate inquiry with more traditional lessons?

Response

The fraction of lessons that are inquiry lessons will depend on how long the department has been using Inquiry Maths prompts. At the start, teachers might run a structured inquiry for one or two lessons only. However, as experience and expertise develop, one inquiry might be extended over more lessons. Ultimately, the department reaches a position in which all lessons, whether they involve independent exploration or an explanation, are all part of an inquiry.

Inquiry prompts should be used at the start of a unit or topic. Through students' contributions about the prompt, which is set just above the current understanding of the class, the teacher can assess the extent to which new procedures and concepts will be required to make progress in the inquiry. The teacher might use 'traditional' instruction to introduce the new knowledge, particularly if students request such an approach through the regulatory cards. However, the teacher is not integrating inquiry with traditional lessons, but integrating a teacher's explanation into inquiry when it is appropriate, relevant, and meaningful.

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