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Covariational reasoning

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  • For teachers and practitioners
  • Espresso
  • Covariational reasoning
  • Espresso
  • 50: Covariational reasoning
  • 49: Teaching and learning equivalence
  • 48: Early development of functional thinking
  • 47: Developing concepts of pattern
  • 46: Building and breaking 2D and 3D shapes
  • 45: Teaching logical reasoning
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30 October 2024

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'.

What does research suggest about covariational reasoning?

  • Covariational reasoning can be explored across the breadth of mathematical activity from early primary onwards
  • Students can be given opportunities to develop covariational reasoning through exploring linear functions and rates; making and critiquing statements; and considering association, correlation and scatter plots
  • Focusing on plotting or comparing pairs of values may limit opportunities to develop covariational reasoning
  • Covariational reasoning in statistical contexts requires students to also coordinate the idea of signal and noise, and this can be challenging
  • It may be beneficial for students developing covariational reasoning to use either time or a counting sequence of numbers as the second variable
  • Exploring covariation through multiple representations is important; students may find it easier to use graphs at first, especially non-standard ones
  • Useful activities might include highlighting fallacies and misconceptions, using modelling, and using real data and/or familiar contexts
  • Software tools can be used to support students’ exploration of covariation by allowing more time to focus on exploring patterns
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