In their early work on measure, students usually position the ruler closely to the object being measured. Visualisation of an orthogonal grid – students equate the length of each bar to a position on the y axis.in a dot plot, or by colouring squares on a grid), but must transition to the idea that the length of the bar itself corresponds to the value of the count. Representing a count by length – students often initially judge the frequency of a data value by counting icons (e.g.Evidence that students either have a weak model of the number line, or do not associate the axis of a graph with the (linear, non-logarithmic) number line can be seen when the spacing between consecutive values are of inconsistent size. The number line – it’s essential for students to understand this model of number in order to be able to read information from the y axis of a graph.Exclusion – once a data value is placed on the graph, an equivalent value cannot be placed directly on top of it.The value of each answer can be represented by the horizontal position, but each response is functionally identical to any student who gave the same response and therefore their own value cannot be distinguished within this set of responses. Equivalence – any two identical responses are equivalent and hence interchangeable.More often than not this icon is an imagined square (or rectangle) whose horizontal position signifies their answer to a question. Correspondence – at the most basic level, students must understand that their individual responses to the question “what is your favourite colour” are represented by an icon (e.g.What concepts are necessary to understand the various non-arbitrary features of such an entity? ![]() Let’s start by focusing on what is generally considered to be one of the most straightforward charts: a bar chart for a small categorical data set such as ‘favourite colour of students in my class’. Students whose broader mathematical knowledge is insecure may be forced to learn graphing rules by rote, rather than understanding their meaning and relationship to the wider mathematical ecosystem. Secondly, because each graph rule seems simple, it is easy to ignore the fairly deep mathematical ideas that underpin each one. What does each axis represent? Where are the individual data values represented? Why is this representation appropriate for this data? Firstly, teaching the mechanics of graph construction encourages students to attend to the ‘how’ without considering the ‘what’, the ‘where’ or the ‘why’ of the different features. Much of this difficulty can be traced back to two specific and related problems. Students seem to come up with a plethora of problems and misconceptions when faced with any task involving graph construction for example unusually inconsistent axes and best fit lines that are in fact quite the opposite. However, maths teachers’ experiences suggest otherwise. It’s tempting to assume that constructing graphs of data is a relatively straightforward task, requiring students to learn a small set of rules for each type of graph, to be used with a multitude of data sets - in some senses an algorithmic approach.
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