The gap... Closed to open / specific to general
In maths lessons without proper planning we can too often allow students to get caught up with chasing down "the" right answer, and therefore get stuck with apparently closed questioning. However maths is a subject that makes leaps from closed questions to open questions, and from specific cases to general cases with huge regularity.
For example we can step from the specific 3+6=9, to the less specific x+6=9 to the general x+y=9, and beyond into the even more general x+y=z. Sometimes we ask students to make this leap in the space of a single lesson, and too often we do it with students that don't fully understand what the letters represent (for more on levels of understanding of algebra see chapter 8, of this book - it's fairly old, but well worth a read if you've not done so before - can provide a real insight into the barriers to understanding algebra). For those of us with a sound grasp of algebra and who are happy to use letters as variables this step from specific to general is fairly trivial. However for those without that grasp it is too often a step that presents a real challenge, or that can be made only by copying procedures without deeper understanding of the concepts that lie underneath.
Vitally though, if we don't plan and structure it properly our questioning can also make these big leaps from closed and specific all the way up to open and general. There is the potential to leave a huge gap in between, where misconceptions and misunderstandings can sneak through without detection.
It was while I was contemplating this gap that I stumbled upon the SOLO taxonomy and realised that this helps me to explain this issue. In terms of SOLO levels of understanding this gap can require a student to take a leap from Prestructural or Unistructural understanding all the way to Relational or Extended Abstract understanding without making any links on the way. (When I found out about SOLO I was amazed that it wasn't part of my basic teacher training; I think it's really powerful. If you've never heard of SOLO before then take a couple of minutes to have a look at this video as it explains all of the key points - also linked here on youtube if this embedded video doesn't work)
Using existing good practice to help fill the gap
I know from observations that there is a range of excellent practice in terms of questioning within the maths department, but even the best of us do leave this gap open at times in our rush towards generalisation. I thought that with some tweaks we could put something in place to help us to bridge the gap.
To capture examples of the existing good practice I asked each member of the department to send me 2 or 3 examples of questions that they had found effective during a particular week. I then took these questions, added a few more, and tried to align them with the SOLO taxonomy. The result is this sheet:
What's the point?
The idea of the sheet is to give us a quick reference sentence starter or framework to build a good question around depending on topic. By aligning the questions with SOLO we can work our way up (or down) the levels of understanding as needed with a particular class or individual. Importantly it structures the questions with several in the Multistructural and Relational areas to help us avoid making jumps too big and losing students in the gap.
Of course those fortunate enough to be really gifted teachers can always from the perfect question at exactly the right time, however for most mortals it can be useful to have the occasional prompt, and there are other uses for a sheet such as this (see below).
Not a definitive list - all about context
This sheet is only intended as a prompt, it's not all encompassing and it should be used with a due level of professional interpretation to judge whether a question is suitable for a given class or student. Similarly the detail of the question can be tweaked as needed, and extended with further questions (ref Pause, pose, pounce, bounce by @TeacherToolkit).
Of course you could debate the location of some of the questions and where I've assigned them in the SOLO taxonomy - this was my first attempt and some are far from clear cut. In fact one of the key points of SOLO is that the response to a question can be at a different level to the question itself. For example the deceptively simple "What is a fraction?" could be Prestructural if we're talking to a student that has only a vague concept of what a fraction is, and it could rise as high as Extended Abstract if we start making links beyond decimals and percentages into algebraic fractions, gradients or differentiation, etc. This is where the context of the question and professional skill/judgement comes into it.
How are we using the sheet?
1) Copies in departmental planners to help form questions at lesson planning stage
2) Copies available in the classroom (soon to be on the wall) to give a quick prompt for the teacher as part of a plenary or mid-lesson review (particularly useful if the lesson hasn't gone exactly as planned so any questions planned in advance can't be used)
3) Whole sheet, or sections of it, given to students as part of a lesson to encourage them to ask challenging questions of themselves or each other as part of group or discussion work.
What's the impact?
Early days really, but having the sheet for reference is undoubtedly useful as part of the planning process. Similarly just asking the department for examples of their good practice provokes reflection and review over and above normal day to day practice. However what is more interesting is where this could take us next...
The sheet is planned to be kept as a live and developing document. I plan to review it in department meetings over the coming 12 months or so and update it with more questions and refinements to make sure it is as useful as possible to the department both for planning and use during lessons.
I also see a route towards questioning structures linked to schemes of work. This link shows an example that @TeacherToolkit has posted using Bloom's Taxonomy to differentiate questioning for a Design Technology topic, and I see no reason why we can't develop a similar approach in Maths using this SOLO structure.
I am also keen to try to embed this approach alongside our work on establishing a common language for feedback (see this earlier post). If the students can start spotting patterns in our questioning and make links to how this helps to develop understanding then I suspect there is a meta-cognitive benefit to be had. However we've not tried that yet so it's nothing more than speculation for now...
All thoughts welcome
As always I'm keen to know your thoughts. Is this useful? Do you have anything similar? Do you have anything completely different that does the same job? Do you think I'm wasting my time (if so - please say why)?
Thanks for reading. Look me up on twitter... @ListerKev