The Glass Wall: Why Mathematics Can Seem Difficult Book

I have been on the wrong side of the glass wall as envisioned by this book. As a learner, mathematics made little sense to me. I couldn't remember the formulas, and I failed the subject in my final year of secondary school in 1981. Ironically, I remember my score of 45%. Now, having been a teacher for over 35 years, I find myself on the other side of that wall. I am gobsmacked that I only became aware of this title late last year in 2019 courtesy of a friend's recommendation. It was published in 2002!

From the first page in this book and you are told that mathematics "isn't necessarily complicated or difficult, nor is it something that is accessible only to an intellectual elite. Mathematics should be open to everyone, provided no unnecessary obstacles are encountered" (p. vii). Frank Smith writes with such clarity about these complicated matters, the obstacles that can keep us behind the glass wall. In this book Smith is writing somewhat out of field, he is known for his work in relation to a whole language approach to the teaching of reading. Perhaps this is why he is able to reveal what makes mathematical understanding possible, and why it can be difficult. The obstacles to understanding mathematics are often rooted in language, how we talk about mathematics and how we write about it. These also reflect how we perceive and teach it.

Rather than go into detail about the book's content, I'd like to share how I responded to reading it. Much of what I read, I read electronically, via my kindle, ipad or computer. However, I read an actual paper copy. And this changed my reading experience. With a pencil in hand, I felt like I was having a conversation with the author. I made notes, comments in the margin, smiley faces even, and successive exclamation marks when I agreed (and I rarely disagreed) with what he was saying. I was taken by his ability to express the complex relationship between language and mathematical operations, for instance, he writes, "Cells multiply by dividing" (p. 33). This lays bare the relationship between multiplication and division, but also the potential for confusion, which is brought to life in the pages of the book. Not understanding this relationship can keep a person on the wrong side of the glass wall denying mathematical understanding.

The book reiterates how language can get in the way. The author at one point discusses what it means to develop number sense and how this is very different from how language develops. He explains that you never hear the phrase 'word sense' or the need to develop 'good word concepts'. That's not a thing.

The Glass Wall emphasises the importance of developing understanding from inside mathematics, from the world of mathematics that minds have created over many, many years. The reader is reminded that "Piaget asserted that children have to invent or reinvent mathematics in order to learn it" (p. 15). I suspect that the world of school mathematics gives little time for such invention to occur. When talking about '4 eggs in the nest' or '3 pencils' to somebody, the idea of fourness and threeness is not obvious, unless you know 'four' and 'three'. As he says, "for as long as we regard something as obvious, we overlook the intelligence that went into making it self-evident" (p. 4). This is similar to how place-value is taught. When we tell the kids, 10 of these is 1 of those, then show them a number perhaps via a picture of an abacus, that, in itself, doesn't even reflect the pattern, then give them a worksheet to complete, what we have actually demonstrated is that we have overlooked the intelligence that makes place value self-evident to us, but perhaps incomprehensible to our students.

Unsurprisingly, a couple of chapters are devoted to considering place-value, that is, how numbers larger than 9 are written and to the names given to such numbers. Smith writes "Numerical order is not found in the physical world, except when we put it there. There is no 'order' in the natural world because there are no numbers there. The notion of order … comes from our fertile brain" (p. 64). I had a conversation with a 16 year old in the course of my work and asked him how he might represent the number sixty-six. His first response of what to write is: '66'. I asked if he could show me another way, to which he responded by writing '60 + 6'. I then asked if he could show me without using numbers. He drew 66 tally marks clustered into chunks of 5 as a way of keeping track of a count of ones from 1 to 66. For him, 66 was a collection of 66 ones. A person aware of the place-value structure would also recognise 66 as 6 tens and 6 ones. When we fail to see the structure in our own mind, as this person had, we come up against the glass wall.

The Glass Wall gives insight into the mathematics beyond counting, that of, calculating, measuring, the numbers that exist between numbers, and geometry, as well as some suggestions for getting beyond the glass wall of incomprehension. My only criticism is how memorising is treated. On the one hand, the author acknowledges that "Memory is organised on the basis of understanding. Everything we remember is connected to something else that we know…" (p. 116). I love this, yet, he then suggests a "minimal tool kit for basic mathematical memory" (p. 117) would have to include the number facts for addition and multiplication up to 10. If these facts are seen as separate entities to be committed to memory, then there are 200 discrete facts to 'learn'. There is an alternative to thinking this way, through teaching strategies that have a 'long shelf life'. For instance, consider 5 multiplied by 9 (5 nines) that can be thought of in terms of knowing 10 nines, 90, and so half of this is 45. I can apply this thinking to 5 multiplied by 23 (5 twenty-threes): thinking 10 twenty-threes, 230, half of which is 115. And this is just one of the strategies that might help a student get us beyond the glass wall.

The book has given me a great deal to think about, especially in terms of the things we humans look for: "consistency, coherence and consensus" (p. 125), and what Smith has identified as the four essential conditions for learning mathematics. One of these conditions is 'time', that is, giving learners the 'gift of time' a phrase I first came across through a friend and colleague who introduced me to Richard Lavoie (see for instance his book, It's so much work to be your friend: Helping the child with learning disabilities find social success). Because, as the author of The Glass Wall says, "learning can't be forced … learned in a hurry" (p. 127). I highly recommend taking the time to read this book.