 Hi my name is Jenny Ingram and I want to invite you to attend to some little words. There is a lot of discussion and talk about vocabulary mathematics teaching but I want us to focus on little words that we often say without noticing them. The words whose meaning makes a difference to the mathematics but perhaps in ways we are not aware of when we use them. Take for example these four statements. The price was reduced to ten pounds. The price was reduced by ten pounds. The price was reduced from ten pounds. The price was reduced ten pounds. As we move from one statement to the next just one word is changed yet that one word completely changes the meaning. Some of the changes might change how we think about a statement. Giving you this statement the square of the hypotenuse is equal to the sum of the squares of the other two sides. Some of you might be thinking about this equation what we call Pythagoras' theorem. Though some of you might have been thinking about this equation or possibly this equation but if we pay attention to the changes in little words in this case of we might begin to think in a different way. Here the square on the hypotenuse is equal to the sum of the squares on the other two sides. I invite you to pause for a moment and notice what is the same and what is different in your reaction to each of these statements. Show me a number divisible by six is even. Show me that all numbers divisible by six are even. Show me that any number divisible by six is even. Can you notice something that was the same? Can you name this thing that is the same? Where you notice something that was different what are the consequences of this difference? For many of these little words their use in different contexts can also influence how we respond to statements. In some cases we might spend some time working with students explicitly on the consequences of these little words on the mathematics that we're doing. For example we often explicitly teach what the word and and or means within probability but in the context of quadratic equations we might not stop to think about whether the choice of word matters. Take for example factorizing quadratic equations to find the solutions. Are these solutions x equals 3 and x equals 2 or are these solutions x equals 3 or x equals 2? Why might different teachers and students argue that the word and is appropriate? Why might different teachers and students argue that the word or is appropriate? Here's another situation. Here I invite you to take some time to think about each of these statements and what we mean by same in each context. So two thirds is the same as six ninths. 54 take away 22 is the same as 58 take away 22 take away 4. Sine 2x is the same as 2x, 2 sine x cos x. 5x plus 2 equals 2x minus 7 is the same as 3x plus 2 equals minus 7 and then for this diagram angle AOC is the same as angle BOD but in what cases the angle AOC the same as EOF? In both those situations we've used the word same but is that sameness the same? So in what way are each of these pairs the same? Are we hiding the mathematics underpinning each of these statements by using this phrase the same as? And to end with I just want to shift our attention away from mathematics briefly towards how we say what we say affects the learning of mathematics. Here are some words I often hear teachers say and also catch myself saying but who is it obvious to? What is it that is obvious? Each of these words implies that there is something our students should know that makes something obvious or easy. What if you're that student who does not know or does not remember in that moment that knowledge that is obvious to you? To end with I invite you to do a couple of tasks. So I invite you to return to each of these statements that we talked about earlier and think about different ways of expressing the meanings and in particular the actions involved in order to say that one is the same as the other. For a second task I also want you to think about the many other little words in mathematics or words that we use but do not necessarily pay attention to such as explain or simplify. Another task for you is to construct some statements like the ones in this slide from a range of mathematical topics within the maths curriculum that use one particular word such as simplify where the meaning varies between these examples.