# Barred for life: Some thoughts about exercise books, rose gardens and bar modelling

Lines, squares or plain? The pages of maths exercise books, I mean. For years I have insisted that pages have to have squares. Without vertical lines as well as horizontal ones how could my students possibly draw all the right-angled triangles and bearing North lines and coordinate axes and so on that I would tell […]

# Building the Maths house: Singapore’s curriculum framework

I thought I’d write today about the framework of Singapore’s School Mathematics Curriculum.  The framework is captured in a well-known diagram that I’ve attached above, and it provoked a lot of interest among teachers when I was last in the UK in November.  This was great to see, because this diagram really is at the […]

# Collaborative Working – the Singapore Approach to Teaching Maths

In November I was in the UK delivering talks on the Singapore approach to teaching secondary mathematics.  While I’ve been delighted to share some of the details of our system during these talks, the real pleasure for me has been to interact with UK teachers and to learn how things are done over here. From […]

# 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 […]

# Original Tangents

I want to offer an example of a mathematical exploration which is likely to enrich learners’ appreciation of an apparently unrelated aspect of graphs of functions. Imagine the graph of . Is there a point on the curve at which the tangent passes through the origin? One thing that emerged when I used this with […]

# Exit Cards

Sometimes we are unsure how well a topic has been received or understood by the class.  We don’t want to wait until they can’t do their homework, which you don’t find out about until they come into your next lesson, to do something about this. Collecting a quick snapshot from every student about their level […]

# Exchanging, Trading and Substituting

Having a vision of mathematics, of what is possible in the way of appreciating and comprehending mathematics (distinct from simply gaining facility with arithmetic), is surely an important part of teaching people to think mathematically. One important aspect of mathematical vision is being aware of ubiquitous themes, and one theme that has been impressing itself […]

# Subbing zero

Consider x + y = 5 How do you know what this will look like plotted on equal aspect axes?  How do your students know? Some possible ways include: using a graphical calculator display rearranging the equation and plotting three points (including one to ‘make sure’) using one intercept and the gradient of –1 recognising […]

# Graph yourself

One of my favourite lessons on graphs is when I use the floor of my classroom as the x–y plane. I fix an origin in the middle of the floor (in order to make sure negative coordinates are included). I use masking tape to represent the axes along the floor, with masking tape dashes marked […]

# More doing & undoing and by-passes

Suppose you want students to gain facility in factorising the difference of two squares, and make use of this when expanding a product of the sum and difference of two quantities. A direct approach might be to invite learners to expand products of sums and differences of quantities, varying the complexity and format of those […]