It has been now just two years since I reviewed Mr Barton’s stellar first book. I say “just,” in part because the last three weeks during this pandemic have felt like five years, and in part because Barton packs so much into his second book, it is a little surprising he did it in just two years.

The central theme of Reflect, Expect, Check, Explain is using and constructing ‘intelligent’ sequences of mathematics exercises, “providing opportunities to think mathematically.” The intelligence behind these sequences is the way we order and arrange them, allowing for comparison (reflection) between two or more exercises, the anticipation of what the answer or solution method will be (expectation) based on what the previous answer or solution method was, determination of the answer (check), and then an explanation of the connection between the exercises (explain).

Consider, for example, the sequence at left, from early in the book. During **reflect**, for the first pair of exercises, I can notice that the lower and upper bounds have stayed the same, and the second number line has minor ticks for every second minor tick of the first number line. I can also notice that the sought-after decimal value is at the same location on both number lines. This noticing can lead me to **expect** that since I identified the missing value for the first number line as 2.6, my answer should be the same for the second number line. It’s possible, though, that I won’t come up with an expectation. In the **check** phase, I fill in the values for the equal intervals on the second number line, coming up with the value for the question mark. Finally, when I **explain**, I either have a chance to talk about my earlier expectation and explain why I was off or why my expectation was correct or, if I couldn’t formulate an expectation, I can explain why the question-marked values are the same even though the tick marks are different.

As I move through the sequence, there are really interesting thoughts to have.

- Why did the question-marked values line up when moving from 10 to 5 equal intervals (between Questions 1 and 2) but not when moving from 5 to 4 equal intervals (between Questions 3 and 4)?
- Why does “lining up” fail me in Questions 4, 5, and 6 when it worked between Questions 1 and 2?
- I can’t rely on inspection every time to figure out the intervals. Is there something I can do to make that task simpler?
- Is the question-marked value in Question 9 just the question-marked value in Question 8, divided by 10?
- Can I extend my interval calculator method to decimals?

If this were the entire book, that would be enough for me, to be honest. But Mr Barton spends an exemplary amount of effort addressing possible questions and misconceptions about such sequences (the FAQ chapter is excellent) and explaining how these sequences can both fit into more extensive learning episodes and can function in different ways from practice. All the while, the sequences remain the stars of the show.

I highly recommend (again) Mr Barton’s book, especially to math teachers. He outlines in brilliant detail how you can turn a set of boring exercises into a powerful method for soliciting students’ mathematical thinking. No revolution required.

Choice Quotes

Below are just a few snips from the book that I added to my notebook while reading. These are not necessarily reflective of the entire argument. But after a long day of educhatter, which more often than not reads like an ancient scroll from some monist cult, it is comforting to read these thoughts and know that there is still a place for practical, technical, dispassionate thinking about teaching and learning in the 21st century—a place for waging the cerebral battle, rather than constantly leading with our chin or our hearts.

Teaching a method in isolation and practising it in isolation is important to develop confidence and competence with that method, and indeed, students can get pretty good pretty quickly. But if we do not then challenge them to decide when they should use that method – and crucially when they should not – we are denying them the opportunity to identify the strategy needed to solve the problem.

There are two main arguments in favour of teaching a particular method before delving into why it works.**The path to flexible knowledge** The key point that Willingham makes is that acquiring inflexible knowledge is a necessary step on the path to developing flexible knowledge. There is no short cut. The ‘why’ is conceptual and abstract. We understand concepts through examples. The ‘how’ generates our students’ experience of examples. In other words, often we have to do things a number of times to appreciate exactly how and why they work.**Motivation** As Garon-Carrier et al (2015) conclude, motivation is likely to be built on a foundation of success, and not the other way around.

The mistake I made for much of my career was trying to fast track my students to this [problem solving] stage. This was partly due to my obsession with differentiation – heaven forbid a child should be in their comfort zone for more than a few seconds – but also based on my belief that problem solving offered some sort of incredible 2-for-1 deal. I thought it would enable my students to practice the basics, whilst at the same time allowing them to develop that magic problem solving skill.

I will again quote John Mason: “It is the ways of thinking that are rich, not the task itself.”