Santa's Problem

Santa's helpers have gone HiTech, and this year as they have read children's letters, they have logged all the children's requests in Excel. Well spreadsheets are great tools for representing data aren't they? But wait.. Why do we create charts? What purposes do they serve? Who are they for? Do all charts work in the same way? And is the way they represent information always useful?

As is fairly typical of a group with a new toy they have published the results in all sorts of pretty formats, but as we know not all of these are terribly useful unless we are aware of their purpose, and the contexts in which they can be used. So here we are, it is Christmas Eve and Santa is now struggling to use the tools he has been given to help him load his sleigh.

Part of being a good mathematician is not just being able to do the math, but is also about reasoning and thinking about which tools and strategies match the task. This Bar Chart was the starting point for this weeks maths sessions with my class. Was this the right tool to help Santa with the jobs he needed to do before setting out on his journey? If not how could we use the information we had to help us devise another tool that would make Santa's job of packing the sleigh easier.

The week was based on a teaching sequence that I hoped would allow students to explore how this information might be transformed and reprepresented, we carried out our own class survey, using a familiar device the pictogram, as a tally chart to help collect our data. We discussed the gifts that we would like for Christmas drawing and labeling these on sticky labels, then coming together to group and sort these on freize paper. We then discussed what we could see, focussing on the idea of this chart as a story. What story was our emerging "tally chart" telling us? Exploring the pictogram as a starting point and data source, we began to identify tools that were missing that might help our reader better understand what we had recorded. There was no title, and as the discussion unfolded an x axis was added, to label the gift categories. This too needed a label so our reader would understand that these were gifts, not random items. Since we had been working on scales, in one form or another for the last couple of weeks, one student suggested that another line could be added up the side and numbers marked in, our chart could have a scale and it could count in ones, (hmm perhaps we could add a y axis?) This scale would also need a label so readers could understand what it meant. The students were encouraged discussed the story again, and this time as the students commented these were recorded on the whiteboard as if responses to questions, before being asked to work in pairs to rewrite these as questions, rehearsing them aloud and remembering to include appropriate punctuation. Both our new hybrid pictogram, student comments and group questions were displayed to support the next session.

During session 2 we returned to the pictogram/tally we had developed yesterday, and the students were introduced to other ways of tallying, we could tick tally, this was abit like our pictogram as a data source, we had to count individual ticks, or images, but another way of representing the data we had collected the day before was to use a five bar gate system, and here we could use "clever counting," counting in groups of 5 and adding remainders. We began by asking how many students were in school yesterday. The children initially guessed, but were reminded of the tool we had used to help us think about gifts for Christmas. How could we use this tool to help us find the answer? If our survey was accurate, it would include one choice of gift from everyone who was here. The process of counting every block as with every tick, is laborious, but we soon agreed that there were 24 students in school. We began to talk about how we could have made counting easier, and I introduced the idea of five bar gate tallying, working through each column on our chart one at a time to build a running total, we had 5 sets of 5 and 4 left over this is 24. To model this system of tallying we carried out another quick but unrelated survey, using Mark Cogan's "Tally chart" ITP, observing what happened, when we reached 5. The students were asked why they thought this system might be called a 5 bar gate, and responses directed us to how the recording looked like a gate, with 4 uprights, and a cross through. Using our data from yesterday the students were encouraged to make their own tally charts, adding a total column. As a class these were then reviewed as we re represented this data, as a class frequency table in Excel, on the IWB. This final step in the session was a preparatory stage for our third session.

Session three involved us in looking at scales, and introducing the y axis. A helpful supporting rhyme for this came from my colleague as "y for the sky," an image which the children seemed to find useful. We began the session with a counting stick, counting horizontally in multiples of two, five and ten, before turning the stick to the vertical position as we had done with other scales last week and carrying out prediction tasks, eg if this is 0 and this is 100, what might each division in our scale be? How do we know? Where might 50 be? What about 20? 30? 25? and so on. 3 different images of yesterday's data were shared in the form of bar charts generated from the class frequency table and discussed. What could we see? What stories were they telling? This lead to children pointing out changes in the way the y axis scales were formed, one was counting in 2s, one 3's and the other 5s. The columns on the chart were all different, "no they weren't... they were different sizes but they had the same numbers in them.." beginning to refer back to previous sessions on scale some of the students began to recognise that even though the appearence of the chart content was different in each case, the scale on the y axis was determining how these numbers would be shown, and rather than each chart representing a different story, this remained the same, while how the chart represented the the story visually changed according to our choice of scale. This lead to a discussion of the importance of axis labels and titles. For the rest of the session the students were given paper charting frames, in which to represent the data we had collected, and challenged to include y axis scales, adding axis labels and good titles that would help their readers, understand the story their chart was telling.

With previous groups of older students I have used Excel to do this task, the students working in pairs to create charts and add labels at the computer. Although being pleased with the level of understanding the children generally achieved through this task the final outcomes showed some misunderstandings, not around charts, the visual representation was fine, but the paper based tool I had given for individual work. The label frames on the y axis seeming to cause particular problems, for some students who used this not to add labels, but to include their scales. On reflection, I think when I do this again, I will organise the group differently, having them work on the paper frame in pairs, and encouraging talking twos. We also need to consider the placement of numerical values on the y axis, we have not done any number track work this term, but student experiences, lead them not to mark intervals in some cases on the scale divisions, but in the spaces between, which may have been a confusion with how the x axis was organised on this chart type, or the tool they were offered, and is something I need to revisit and address in one more session. Although useful as an exercise in applying what they knew, it might have been better to build up the chart in stages, beginning with the addition of the scale first, moving to titles and labels as a class, before allowing the students to add the data.

Our final stage of the task was to ask which if any of these charts would be helpful to Santa when he began to load his sleigh. The discussion was great and the conclusions of the class resounding, all of the charts help him load it, but would be of little use when he came to deliver. He would need something more. Getting the gifts on the sleigh at one end was fine, but how would he know where to deliver them, and who was to get what. He could use a tally chart, or a frequency table with a check list/tick box to ensure that he had the right number of each gift, but to be extra safe he would need a list of addresses so he could unload at the children's houses when he got there.

Number play and exploration of the relationships between number and data, during my research project literature review, has been shown as something we rarely get to as an inherent part of data handling activity. A growing body of research shows that students involved in data handling activity, frequently spend more time drawing and colouring in graphs, than they do engaging with them to support questioning and reasoning and so rarely have time to use these for their intended purposes, as problem solving tools. The discussion, questioning and reasoning activity that began to emerge from this series of tasks was really useful in laying the foundations for future work, and I am looking forward to further work later, building on these foundations when we use data handling software together to engage with investigational activities around other themes later.