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What does a good maths book look like?

by Cordelia Myers, Nov 2018

A group of secondary Head teachers asked me to consider “What does an effective maths lesson look like?” (a fascinating topic to think about). They then asked: “What does a good maths exercise book look like?” I have found this difficult. I have read around the topic, thought about it and asked a variety of respected colleagues. There is very little consensus.

“It should contain a set of notes so they can revise”

“Most students buy revision guides so there is no point wasting time writing out notes”

“They should show evidence of thinking”

“They should show a journey of learning, with reflections and corrections and annotations”

“It’s a place to process”

After having discussed this question, a number of colleagues have approached me (outside of the meeting) and whispered (as if in a confessional) “I really don’t care what their book looks like”. By this, they don’t mean that students can treat their book with disrespect. None of us want that. What I think they (and I, if I’m honest) want is for an exercise book to be a place where students can express their thinking and, if it’s a bit messy, that really doesn’t matter.

A book reflects lessons. If we return to “What does an effective lesson look like?” we focused on students engaging in mathematical thinking and problem solving. These activities need to be evident in a book.

Mistakes are key. If there aren’t any, perhaps the teaching is pitched incorrectly. Similarly if the book is full of errors. Students’ responses to mistakes are interesting. Are there annotations showing they understand the error or is it always just crossed out and replaced with the correct answer? Is there evidence later in the book of misconceptions put right?

I like to see annotations in a book. Most teachers will present various methods and will encourage students to consider alternative routes. I would expect to see evidence of some of this. For example: Today with my year 8s we looked at 0.003 x 60. The first student suggested we calculated 0.03 x 6, someone else 3 x 60 ÷1000, then 3 x 6 ÷ 100 and so on. You could hear the “I like that”s whispered as they listened to each other. Recording their preferred route helps to consolidate it. What I’m trying to say is that in a good maths book you might see solutions and alongside them other solutions (to the same question) that are mathematically more elegant or efficient. This shows progression of thinking.

In most classes there will be a student who tries to avoid thinking by writing copious notes. There will also be students who spend their time thinking and refuse to write any words. Of the two, I would prefer the latter. Students learn in very different ways and I think, therefore, that their books may look different. If they don’t you would worry about how formulaic the teaching is. Are they simply copying from the board/text book/their neighbour?

I think I have to add: Most maths classrooms will have a student who finds writing difficult. Their work may be all over the place. Maths can be a subject where they shine. We will try to help them establish some structure in their working but if it is messy, that’s not an issue. You can almost hear the student waiting for you to rebuke them (perhaps they are aware and ashamed?) and when it doesn’t happen they often flourish. These are frequently the students who achieve very high progress in maths in comparison to other subjects.

Books, like students and their teachers, have many ways of being “good”. A good book will show a balance between achieving and learning by the ticks and crosses, by the annotations, by the increasingly complex “workings out”. It will show progression of thinking.

There isn’t a simple answer to the question or, in my opinion, a one-size-fits-all solution. Given the diversity of teacher opinion and the diversity of learning styles I wonder if a sensible way forward is for departmental discussions to take place. “Who is a book for?” being a good starting point. Notes reflecting the diversity of views would reduce pressure for uniformity and teachers can encourage students to use their books in the way that best facilitates thinking.