# A reply to Mr Barton

Earlier this year Craig Barton's book How I Wish I'd Taught Maths appeared. Many colleagues told me of their disappointment that Craig (the TES adviser) had decided to listen to a limited field of researchers. A few teachers have even expressed dismay that they had booked to hear Craig talk at one of his frequent conference appearances only to feel deceived by his turn towards cognitive science. I began to hear that Inquiry Maths figures in the pages of the book. With a sense of foreboding I recently got hold of a copy to see what Craig had written. Unfortunately, there is no index in the book, which seems a curious omission for a book with pretensions to synthesise academic research. The references to Inquiry Maths - three in total as far as I can see - come at the beginning of chapter 3 on explicit instruction. It is this part of the book and, in particular, section 3.3 ('When and why less guidance does not work') that I will address in this post.

It seems that the two Mr Barton podcasts I have been invited to do (the first in 2013 and the second in 2017) bookend a transformation in Craig's thinking. Of course, everyone has the right to change their mind, although I cannot think of such a dramatic volte-face as the one Craig delivers. However, in the style of a zealous new convert, Craig has a one-sided view of history.

He relates in the book how he used the surds prompt (pp. 97-98). The example is curious in respect to Craig's conversion to explicit instruction. He showed his students the prompt at the end of a unit on surds that presumably involved some of the explicit instruction that Craig now extols. The fact that the students' insights and conclusions were "rarely complete and often erroneous" might say more about the effectiveness of the instruction than it does about inquiry. I would have used the prompt at the beginning of the unit to develop curiosity and questioning. Instruction is then more meaningful because the students have helped to create the context in which it is given.

However, the main point I wish to make about the re-writing of history concerns how Craig evaluates the use of the prompt. He claims that the students' discussion and debate and the teacher's corrections and re-explanations wasted time, concluding, rather dramatically, that "it broke my heart." Now, in June 2013 (at the time of the first podcast) Craig's evaluation of using the simultaneous equations prompt was completely the opposite. Even allowing for the hyperbole that goes with publicising a podcast, Craig's tweet (below) expresses his enthusiasm for Inquiry Maths.

Unsurprisingly, this does not make it into the book. The reader would have had a more honest picture of Craig's past engagement with Inquiry Maths if it had done.

Before turning to Craig's critique of inquiry, let's get one thing clear that a reader of How I Wish I'd Taught Maths might find hard to believe. Research supports inquiry learning in mathematics. Blazar (2015) concluded that inquiry-oriented instruction is positively related to student outcomes, which "lends support to decades worth of reform to refocus mathematics instruction toward inquiry and concept-based teaching" (p. 27); Bruder and Prescott (2013) report that "the effects of IBL include benefits for motivation, for better understanding of mathematics, and for the development of beliefs about mathematics as well as for the relevance of mathematics for life and society" (p. 819); and Lazonder and Harmsen's meta-analysis (2016) starts by saying that "research has consistently shown that inquiry-based learning can be more effective than other, more expository instructional approaches as long as students are supported adequately" (p. 1).

The final point, which I have placed in italics, is an important one. Inquiry involves different levels of structure and guidance depending on the experience of the class. For Craig, this seems tantamount to cheating. He writes, "Now, of course, we can offer our students support and guidance to help them approach these tasks in a structured and systematic way, but by doing so are we not moving further towards a more teacher-led form of explicit instruction?" (p. 101). My initial response to this question was, "Yes and so what?" However, as I consider it further, the sentence expresses a highly damaging dichotomous view of maths teaching. Either you do inquiry or you use explicit instruct. Apparently, borrowing from the 'other side' negates your approach.

Just this week, a new colleague observed me running an inquiry with a year 7 class. He likened parts of the lesson to explicit instruction. Should I be upset that our inquiry descended into such evils? Certainly not. The context of the instruction had been established by students' questions and observations about the prompt and the situation called for an explanation of indices.

Craig replies that inquiry remains a "less guided approach" that places "greater responsibility in the hands of the student" and leaves students "free to pursue their own paths and lines of inquiry" (p. 100). Yes, but open inquiry in which students are free to choose their own approach is the aim of the inquiry teacher, not the means by which it is achieved. You do not teach students to conduct mathematical inquiry by throwing them into open inquiry.

The final point I wish to make is a more general one about our aims. Craig acknowledges that "students can and do learn basic skills from these [inquiry] activities, but it will take a lot longer, and with the likelihood of many bumps and bruises along the way" (p. 106). And there's the key. Experiencing bumps and bruises is exactly where our students learn what it means to learn and reason. Even if cognitive scientists try to convince us that teaching can be made 'efficient', the emotional and social aspects mean learning is far more nuanced than the sterile, programmed classrooms envisaged in How I Wish I'd Taught Maths.

Furthermore, maths teachers should be in the business of doing more than overseeing the acquisition of basic skills. If we go back to Craig's tweet, we find the emotional and social that he has stripped out of teaching. "Questions and theories were flying," he says. Teachers know how exciting a lesson is when students are engaged as mathematicians in a collaborative search to answer their own questions or develop their own conjectures. These are the classrooms in which inquiry interweaves the learning of basic skills with the big picture of connecting mathematical concepts.

Craig fears that inquiry is "a bad thing." Well, the most innovative and courageous maths teachers I know use Inquiry Maths prompts regularly. Some teach in international and independent schools; others, like my colleagues and I, teach in comprehensives serving deprived areas in which the vast majority of students are entitled to free school meals. Some teach students who have the highest attainment of their age group; others teach students who have developed a negative attitude towards mathematics. They use their professional judgement to introduce students to the excitement of inquiry. That, Craig, is a wholly good thing.

Andrew Blair, October 2018