Scenario: Jim teaching construction of bar charts from tally charts

Country: UK

Grade (student age): Year 6 (age 9-11)

Contributed by: Fay Turner, University of Cambridge, UK

Context – national, curricular, professional, other

Jim was a student teacher in the final term of a postgraduate teacher education course. During his final school placement, he was teaching a unit about data handling to a Year 6 class.  In a previous lesson he had shown them how to make tally charts to record the frequency of different phenomena. One of the displays in the classroom consisted of a number of different bar charts that the pupils had made previously.  At this time, curriculum guidance for mathematics in England suggested that pupils should be able to use tally charts by Year 4 and to construct and interpret bar charts by Year 5.  Learning objectives for data handling at Year 6 included ‘solving problems by representing, extracting and interpreting data’ and suggested that pupils should be able to understand bar charts with grouped discrete data’.

Scenario

In the introduction to the lesson Jim showed the class a tally chart indicating numbers of cars of different colours that had passed his window and then showed a bar chart representing the same data.  Jim asked a number of questions about the tally chart and bar chart before introducing the pupils’ tasks.  These tasks involved making bar charts representing the number of pupils with different numbers of letters in their names.   The lower achieving children were given tally charts from which to construct the bar chart.  The higher achievers were given lists of names and told to use these to make the tally chart before constructing the bar charts.  All pupils were asked to set questions for others to answer once they had completed the bar charts.

Most children in the class appeared to struggle with these tasks.  The lower achieving children, who were given the tally chart, could not decide on which axis to put ‘the number of letters in names’ and ‘the number of tallies for each number’.  The teaching assistant working with one lower achieving group modelled putting the ‘number of letters in names’ along the horizontal axis starting with zero. Jim explained to her that the numbers of letters should be seen as ‘categories rather than numbers’ and therefore there was no need to start from zero on the horizontal axis.  This demonstrated an ‘overt display of subject knowledge’ (OSK) another of the codes of the Knowledge Quartet.

In his explanation of the tallying task for the higher achieving children, Jim modelled counting the number of letters in two of the names on the list.  He wrote the number of letters on the board along with one tally mark to indicate that one person had that number of letters in their name.  Jim told the children that they needed to keep tallies for each ‘number of letters in a name’.   The children appeared to find this confusing and several of the higher achieving children put the names themselves rather than ‘the numbers of letters in the names’ on the horizontal axis of their bar charts.

Realising the difficulties pupils were encountering, Jim interrupted the activity and modelled the construction of a tally chart on the board.  Jim reminded the children of the need to “count the number of letters in each name and put a tally next to each number”.  He drew a table with the two headings ‘number of letters’ and ‘total’.  Joe asked several pupils for the number of letters in their names writing the number under the first heading and a tally mark under the second.  Joe suggested that his name and the teaching assistant’s name each had the same number of letters as names of pupils he had recorded. Joe put a second tally against 11 and 20  in the table to demonstrate tallying the number of people with the same number of letters in their names.

Number of letters [in names]	Total
11	I I
7	I
15	I
12	I
20	I I

 

No children in the class completed the bar chart representing the frequency of numbers of letters in names.  They were not therefore able to use these to ask questions.  For the plenary, Jim returned to the bar chart showing numbers of different coloured cars about which he asked some questions such as “How many yellow cars passed my window?”

 

Knowledge Quartet Coding Commentary

Contributed by: Fay Turner, University of Cambridge, UK

Knowledge Quartet Dimension: Foundation

Knowledge Quartet Code: Awareness of purpose (AP)

Scenario: Jim teaching construction of bar charts from tally charts

Jim agreed that this lesson had not gone well.  He belatedly realised that finding the frequency of occurrence of numbers of letters in names involved a greater level of complexity than finding the frequency of different coloured cars.  This lesson might therefore be used as an example to which the KQ code ‘anticipation of complexity’ (AC) should be applied.  However, the overarching difficulty with this lesson might be attributed to a lack of ‘awareness of purpose’ (AP).

By the time they reach Year 5/6 pupils are likely to have experienced constructing bar charts a number of times and the work displayed on the walls of this classroom suggested these pupils had done so recently.  The reason for getting them to construct yet more bar charts therefore seemed unclear. Jim may have wanted the pupils to move to a next step in their understanding of bar charts where each bar is seen to represent not just one option or category, but a range of numbers.  In this case, bars may have represented 3-5 letters, 6-8 letters, 9-11 letters in names and so on.  Neither Jim’s introduction involving colours of cars, nor his explanation of the task, suggested that this was his purpose although the task itself probably did require understanding this next step. A greater awareness of the learning this lesson was trying to promote might have led Jim to consider more carefully how this purpose might be achieved.  This would have led to focusing on the range of numbers of letters represented by each bar and helped pupils understand the tallying task.

There is a second sense in which awareness of purpose appeared missing from this lesson. The use of realistic contexts for mathematical problems help to give purpose to the teaching and learning of mathematics as well as helping to support pupils in carrying out tasks with understanding (van den Heuval-Panhuizen, 2001).   Data handling may be considered to be the area of primary mathematics that can most easily be taught within meaningful contexts.  However, in this lesson pupils were never given a purpose for keeping tallies or representing information in the form of bar charts.  Jim presented a tally chart and bar chart of the number of different coloured cars he purported to have observed going past his window.  However he did not attempt to give an explanation as to why he might have collected this information.  Similarly, the pupils were not given any reason for wanting to record information about the number of pupils with different numbers of letters in their names.  It is possible to think of an explanation, e.g. the need to know the amount of tape needed when making ‘ticker tape’ labels for drawers.   However, more realistic scenarios might have been found that would have facilitated development of the same skills.

In summary, this is an example of AP code because these were two senses in which Jim did not appear to be aware of the purpose of the lesson.  Firstly, the construction of tally charts and of bar charts served no purpose in extending the pupils’ learning.  Secondly, Jim did not attempt to present a reason for the task, i.e., finding how many children had different numbers of letters in their names.  An explanation for this is difficult to find and he might have chosen to collect and present different data for which a more realistic explanation could be given.  The pupils were presented with a task to complete rather than a relevant question to be answered by the collection and representation of data.  It is such a lack of purpose that leads pupils to believe that the only reason for learning mathematics is in order to successfully complete tasks in mathematics lessons.

References

van den Heuval-Panhuizen, M. (2001)  ‘Realistic mathematics education in the Netherlands’, in J. Anghileri (ed.) Principles and practices in arithmetic teaching: Innovative approaches for the primary classroom (pp. 49-63).  Buckingham: Open University Press