In researching available content for MathHooks.com, I came across the site Inquirymaths.com, which is a fascinating site and shares many of the aspirations that I have for math class. In short, the site is a home for material and discussion for an inquiry approach to math. The site is quite dense with ideas and innovations but to summarize as briefly as possible, inquiry maths relies on a single "prompt" which is often (and ideally) as simple as a diagram. At times, the diagram is accompanied by a single statement or question. A successful prompt should do all of the following:
1.) Promote curiosity
2.) Aim to develop mathematical knowledge
3.) Be accessible to students but just beyond their recognition
4.) Prompt should be able to be manipulated and/or changed
5.) Open to a number of pathways of inquiry
6.) Offer opportunities for both induction (explore and generalize) and deduction (logic and proof).
7.) Abstract (not real world)
Here's an example of an inquirymaths.com prompt.
A typical prompt in inquiry maths will allow students to explore anywhere from 30 minutes to an hour and has the potential to lead them wherever their mathematical knowledge wanders. This, in my opinion, is very cool! But this is also where math hooks and inquiry maths divide and go on their own separate paths. Inquiry maths is meant to give students the opportunity to investigate and explore their own math ideas without boundaries or limitations. MathHooks' aim is to build discussion with the explicit goal of constructing ideas about a specific standard, which inquiry maths advises against when creating prompts. I point out this difference not to argue the value of one approach over the other but rather to highlight the differences in the goals of each approach.
Another interesting aspect of inquiry maths that I found was the idea of creating problems (prompts) that are just beyond students mathematical reach. Michael Fenton, a teacher from Fresno California, offered the idea of using low floors for prompts on his blog. As a big fan of Dan Meyer, I immediately recognized this as an ideal. However, Andrew Blair of inquirymaths made a solid argument to create problems just above students level of recognition and cited Vygotsky's Zone of Proximal Development to support his argument. Because my own background has always been in urban settings where students often enter class a grade or two (or three...) behind, I like the low floor. However, in creating math hooks aimed to grade level standards, I find myself adhering much more to Blair's ideas of setting hooks just above students' recognition. Of course, I think the approach a teacher should take depends largely on their own judgement of which entry level works best for the specific lesson they are teaching. In either case, the goal is the same: rigorous inquiry of mathematical ideas.