Inquiry maths reviewed
On this page, we reproduce two blog posts that review inquiry maths. The posts were written by University of Brighton students who are on a Subject Knowledge Enhancement course before starting the PGCE (a UK teacher qualification) in mathematics. Set nine tasks by their tutor Peps McCrea, the students are invited in the fourth task - Critique and Inquiry - to reflect on a video about the Khan Academy and an article on inquiry maths. The first post*, by Alison M, asks how long students should be left to struggle in open inquiry. The second from Ben raises the concern that inquiry is yet another "one size fits all" approach to teaching. Andrew Blair replies to both posts.
* The second part of the post is reproduced here.
Moving onto Andrew Blair’s Inquiry Maths article, this too lends itself more to pupils being able to explore solutions for problems independently and become more engaged in the learning process. Whilst I am in favour of the idea of “Inquiry Maths” lessons, the concept is very reliant on the pupil’s ability to think independently and critically and for those pupils who struggle with discovering theorems and proofs they will require assistance from the class teacher in order for them to achieve their “light bulb” moments. However, Inquiry Maths lessons would lend themselves to greater shared learning and the development of a wider portfolio of mathematical solutions which would be a great thing.
My enlightenment in maths has generally happened in group discussions whereupon you are able to make conjectures in order to clarify your mathematical conceptual thinking and learning, then when this successfully under your belt you are able to explore the concepts more fully outside class to embed this thinking and learning. I have always found that by asking “What if?” a good basis to explore hypotheses and conjectures in order to understand fully the concepts taught in class, for me this is fundamental in learning new subjects and their proofs – WHY ? HOW? And then BECAUSE…… This allows for us as students now and teachers in the future to complement the concept and method for successful learning. I for one would like to use Inquiry Maths when I teach.
Questions: How integrated is Inquiry Maths in UK schools and how long do we allow pupils to “struggle’’ to solve problems before giving a solution?
Andrew Blair replies: Hi Alison, I greatly enjoyed your piece and I also think the “what if?” question is very important. If students have the knowledge and confidence to transform a mathematical object by asking “what if?”, then they are thinking like a mathematician would think. That, for me, is the point of inquiry maths. You’ve made another important point when you suggest that the concept to be learnt and the method of learning that concept should “complement” each other. Inquiry maths gives students a role in establishing the complementary state. They have the opportunity, through the regulatory cards, to decide what they require to make progress – whether, for example, it is instruction in a procedure or a period of individual exploration.
I’ll respond to your questions in reverse order. I’m not sure the idea of “struggle” captures the processes in inquiry. I prefer to see it as a joint exploration where we (teacher and students) co-construct an understanding of the mathematics buried in the prompt. In this sense, “struggle” would be overcome through a joint responsibility for class enlightenment. It is, however, the responsibility of the teacher to judge the level of inquiry (open, guided, or structured) appropriate for the class. How integrated is inquiry maths in UK schools? One of the three aims of the new KS3 curriculum is to “reason mathematically by following a line of enquiry”, so maths departments should be including some form of inquiry in lessons. To my knowledge, inquiry maths has been used in at least 100 secondary schools and some primaries in the UK, and possibly in 100 more schools abroad. As all materials, ideas, and classroom experiences on the website are open access, it is difficult to say with accuracy.
One size fits half; the other size doesn't fit at all
“One size fits all” is the idea we’re trying to move away from in teaching Mathematics. Yet it seems our answer to doing this is just trying on another massive jumper that’s expected to fit everyone. But it doesn’t.
Andrew Blair’s article on Maths inquiry and Pershan’s critique video are illuminating and clearly demonstrate an excellent way for students to understand Mathematics in what Skemp calls a ‘relational’ way via conceptual learning. This idea that students should be pushed towards asking their own questions and using what they know to find answers is exactly what Maths is about! The question though is will this work for everyone?
Initially I thought; “well this is stupid – how are GCSE students supposed to be able to direct their own learning at that age?” Then I realised that actually doing this from a younger age would be brilliant as it could really help in further education. I was never persuaded to think for myself in Maths and so A-level and degree Maths constantly knocked me down. Practicing this from a younger age is fantastic because their development is so important – not just in Maths but in their ability to solve all problems that come their way in life. This is a crucial skill that should not be abandoned.
My issue with Pershan’s and Blair’s concept though, is that they want to move from one ‘one size fits all’ method of the teacher giving the introduction, method and answer in lessons, to another ‘one size fits all’ method of giving the students a problem and leaving it to them to try and solve it before bringing it all together at the end. Will this really work for all students? Maybe if we always had classes of keen active learners, yes. But the awful truth is that there are often students who struggle to find motivation or enjoyment from Maths. It is therefore possible that these students could find themselves even less motivated when given a problem and left on their own to try and solve it if they don’t know how. Even worse they could hinder the progress of other students in group-work activities.
Maybe the answer is somewhere in the middle. What if teachers introduced an area of Mathematics, then took students’ ideas or questions to give an interactive walkthrough of the problem? Then, the teacher could raise other questions stemming from this problem, which both test and build upon the students’ conceptual Mathematical understanding.
I feel the same method could be used for online Maths videos. I disagree with Pershan’s view that videos should be separated between question and answer because you could always just use that magical ‘pause’ button which does the same thing. But the idea of separating videos into question/answer, then extended activities for conceptual understanding would be a great idea!
In a world where everyone is different – how would you try to teach for everyone?
Andrew Blair replies: Hi Ben, Your piece raises a very important question: how can we design teaching methods that accommodate the needs of all students (or as many as possible) in the class? There is always the danger, as you say in your thought-provoking piece, of turning our preferred style into a “one size fits all” approach. Do we use instruct and practice, let students discover, or take some kind of middle path? The problem for me with all of these approaches is that the teacher decides.
Inquiry maths is different because it provides a mechanism for students to play a role in that decision. Through the regulatory cards, they can decide if, for example, they need instruction or to practice a procedure, or whether they’d like to explore one of their own questions. The skill of the inquiry teacher is to blend the students’ ideas into a coherent classroom experience. Depending on the cards chosen, the teacher might opt for a whole-class episode of instruction, although instruction could involve just a small group or be accomplished by peer-to-peer tutoring. Other groups or individuals would be involved in different activities at the same time. Finally, on your point about motivation, my experience is that students engage with learning at far higher levels in inquiry compared to their ‘normal’ lessons. When students are attempting to answer their own questions and are given a role in directing the lesson, motivation rises in all types of classes. Thank you for reviewing inquiry maths. I've enjoyed thinking about the points you raise.