# Tanks, goats and buses- Anne Watson

Consistent use of images supports the understanding of mathematical structures. The outstanding examples of this in mathematics have become ‘canonical’, that is part of the mathematical canon. At school level these canonical images are: number line; function graphs (thankyou Descartes); 2-dimensional combination grids (thankyou Omar Khayyam and Cayley); Venn diagrams (thankyou Cantor and Charles Dodgson).  […]

# Handrails – Anne Watson

I have been floating the idea of treating key mathematics ideas as handrails for teachers and students.  Recently several positive comments about this perspective have floated back to me, so I am sharing the idea with you. In mathematics, teaching key ideas could be handrails. Handrails can be held, used as guides or supports, or […]

# Euclid and Sherwood – Anne Watson

When Euclid wrote about ratios of lengths and areas and similarity, without algebra, theorems were dependent on spatial representations. Five diagrams, all related to each other, appear in his text in various places, so I designed a ‘Match the theorem’ task in the manner of Malcolm Swan’s tasks.  There isn’t a one-to-one correspondence and you […]

# Exponentiation by Anne Watson

Exponentiation I was prompted by a recent tweet to think more about exponentiation. The problem being posed was how to convince students that the laws of indices apply even when the exponent is not a whole positive number. There is a deep issue lurking here, which is that the more abstract the mathematics, the less […]

Professor Anne Watson has two mathematics degrees and a DPhil in Mathematics Education, and is a Fellow of the Institute for Mathematics and its Applications.  She taught mathematics in challenging schools for thirteen years before becoming a teacher educator and researcher at the University of Oxford.  For most of those years she used a problem-based […]

By Anne Watson Why am I talking about linear equations?  Thinking about them takes me to several important hidden threads in school mathematics, wondering how learners might pull ideas together meaningfully. It is also helpful that there has been some research about ‘solving’ them. What is linear anyway? I started by searching textbooks and the […]

# 10 things to say about ratio?

I was surprised to hear that in the pre-2013 Key Stage 3 national curriculum there was only one statement about ratio, whereas in the current version there are ten statements. I was surprised because I did not remember, from my work on the development of the current curriculum, that there were so many, so I […]

# Deep Mastery

Like almost everyone else in Maths education, I have recently decided to broach the ‘mastery’ word.  It does feel a bit odd to do so, given that it’s been around at least since at least the 60’s, and I’ve never felt a need to use it before – despite having been an advocate of the […]

# What do students know about functions?

First, a bit of theory about learning.  In 1980, Shlomo Vinner, an Israeli mathematics educator, coined the terms ‘concept image’ and ‘concept definition’ to highlight the difference between what students actually know about a concept and the formal meaning of the concept. David Tall, an English mathematician and educator, is credited with promulgating the term […]

# Partially Fractious

Even now I remember how odd it felt when, at A-level, we had to find partial fractions. Instead of solving for x, we had to solve for the numerators of the partial fractions. I don’t remember it ever being explained that we were finding coefficients, the parameters of an equivalent way to write the given […]