As promised…I am fairly sure that I could find more, and if I do I will post them, but these are a good place to start:

•  How Can We Teach Math To Encourage ‘Patient Problem Solving’?  (by journalist Katrina Schwartz on the KQED news website) led me to Dan Meyer’s excellent TEDx Talk, Math class needs a makeover. Dan explains with great passion and skill both why inquiry is important in Maths, but also how to go about reshaping your class around inquiry starting from traditional problem sets and working backwards to something more interesting. This is definitely one to watch, and one to recommend to your Maths Department colleagues.
• Dan Meyer’s excellent blog contains a wealth of ideas and suggestions. I was particularly taken with the Three-Act Math section. It is also worth noting that he freely shares his resources under Creative Commons Licences.
• How inquiry-based learning can work in a math classroom from The University of Texas at Arlington is a much more general article (but is actually what led me to the KQED article and thence to Dan Meyer).
• The Inquiry Maths website contains large number of examples of inquiry prompts, suggestions on how to run inquiry in Maths and articles about inquiry in Maths. I was particularly interested in the blog posts The differences between investigations and inquiries and Levels of inquiry (which is strikingly similar to Callison’s (2015) description of inquiry as “Controlled, Guided, Modelled and Free”) – in fact I would recommend looking at the blog as a whole.

Something important to point out here is that there is perhaps a subtle difference between “Inquiry-Based Learning” (IBL) (particularly in Science and Maths) and “inquiry” as pedagogical approaches. Kuhlthau, Maniotes and Caspari define of inquiry as:

“an approach to learning whereby students find and use a variety of sources of information and ideas to increase their understanding of a problem, topic or issue. It requires more from them than simply answering questions or getting a right answer. It espouses investigation, exploration, search, quest, research, pursuit and study.” (2007, p. 2)

But Buchanan, Harlan, Bruce and Edwards, in their review of the literature on “Inquiry Based Learning Models, Information Literacy, and Student Engagement” say:

“IBL is characterized by some or all of the following key components: 1) a driving question 2) authentic, situated inquiry 3) learner ownership of the problem 4) teacher-support, not teacher-direction and 5) artifact creation” (2016, p. 9)

Notice that “use a variety of sources of information” does not appear in this list, and I think that is another part of the problem. We sometimes think that consulting secondary sources is a vital component of inquiry and that Maths and Science investigations which rely on some form of ‘experimentation’ and don’t leave the lab don’t constitute true inquiry. However, we might be happy with a History inquiry that worked solely with primary sources. Do we need to start to view experimental data in Maths and Science as the equivalent of primary sources in History? It is striking that earlier in the same paragraph Buchanan et al say:

“Despite the fact that IBL has been in the school library world for more than forty years…, IBL is most commonly implemented in science and math classrooms” (2016, p.9)

I would say that in my experience we have almost the opposite problem here – most of my colleagues view inquiry as the natural territory of the humanities. Given the opposite is the case in the States, does it say more about what we consider to be inquiry than about a genuine problem in Science and Maths, if we are having a problem in the UK understanding how to ‘do inquiry’ in Maths?

As a final thought, returning (eventually) to your original question Elizabeth, I would regard FOSIL as an ideal support structure for inquiries of this type – we might just need to think more creatively about the activities that happen during the Investigate stage. While students may still need to consult secondary sources, they may not need to do this in every inquiry – and their ‘experimental data’ is a perfectly valid form of Investigation. As a Librarian I still feel I have a lot to offer in supporting this type of inquiry because my expertise is in the inquiry process itself, and not limited to helping students to locate, evaluate and use resources.