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Espresso A small but intense draught of filtered research on mathematics education, expressly designed with teachers in mind.

The infographic is titled 'development of students' covariational reasoning' and shows three stages of the development of covariational reasoning through the example of listening to or reading books over time. At each stage a speech bubble contains text that might suggest what a student could be thinking at that stage. In the first row the student thinks about each book and the time taken to read or listen to them. The text in the speech bubble is 'I can think about how many books I read or listen to over the year. I think about a book, then the time it took, then about another book and how long that took. I can think about the total number of books and also the total time it took.' In the middle row, the student thinks about each book having a position on a timeline, suggesting they are beginning to think about books being listened to or read over time. The text in the speech bubble is 'I can think about time as going forward in a straight line even when it is not in my thoughts. I can start to visualise the idea that I read or listen to a certain number of books every week, or every month, or every year.' In the third row, the student begins to think about the relationship between the books and time as a rate. The coordination of the two variables is shown as a time-series graph - a line graph with time on the horizontal axis and books read or listened to on the vertical axis. The line representing the rate is curved, demonstrating that the rate is not constant. The text in the speech bubble is 'I can imagine time happening alongside me reading or listening to books, and I know that every time I measure one of these things, the other also has a value. At any moment, I can recognise a multiplicative relationship between them, represented by my 'reading rate'.' The students are shown accessing books via reading, listening and using Braille. The caption reads 'adapted from ideas in Thompson and Carlson, 2017'. Issue 50: Covariational reasoning

An infographic about use models of equivalence, utilising: a see-saw balance, a hanging balance and the number line. These are each shown three times in the state of not equivalent, equivalent and still equivalent Issue 49: Teaching and learning equivalence

An infographic about transforming questions into functional thinking tasks, based around multilink cubes Issue 48: Early development of functional thinking

An infographic showing Different ways of seeing the same pattern sequence Issue 47: Developing concepts of pattern

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