Sessions took place in a familiar room adjoining one of the classrooms. The interviewer7 had spent a day in class with the children previously. Sessions lasted between ten and twenty minutes and were conducted individually.
All children were presented with 12 addition problems followed by three partitioning problems. This fixed order was chosen so that children could begin with familiar addition problems before progressing to the unfamiliar partitioning problems.
Before starting the problems, the interviewer put the materials in front of children in the Physical or Pictorial conditions. The physical representation consisted of 20 randomly placed black Unifix cubes (2cm plastic cubes that can be adjoined linearly - Figure 2.1a).
The pictorial representations consisted of 20 grey squares randomly located on a sheet of laminated paper (Figure 2.1b). A marker pen and board rubber was also provided.
7 The interviewer for all studies reported in this thesis was the Doctoral candidate and a qualified infant teacher
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Figure 2.1: a) Physical and b) Pictorial materials used in Study 1
2.2.3.1 Addition problems
The addition problems were presented using laminated cards. Each card had a number problem written in the format ‗a + b =__‘ using black size 72 Ariel font. Before reading out the sum, the interviewer would present the materials and then ask children to use “the cubes/the squares/your fingers if they helped.” The questions were presented in the same order and consisted of four blocks. The blocks varied according to two factors: Total: total less than 10/total more than 10, and Addend order: biggest addend first/biggest addend second.
The first six questions had a larger first addend with a total less than 10 for the first three questions and more than 10 for the second. The second six questions had a smaller first addend with a total less than 10 for the first three questions and more than 10 for the second. This fixed order of questions was intended to represent questions of increasing difficulty. The questions and their order are shown in Appendix A.
Children were given ten seconds to answer each question. If there was no answer, the interviewer would ask the child if they he/she were still thinking. Children in the Physical and Pictorial condition were told not to use their fingers. If any children did start using fingers in the Physical or Pictorial condition, the interviewer would remind them not to for now, also reminding them that they could use the cubes/paper if it helped. The problem would end if the child gave an answer, said they were not thinking
122 any more, or on the third wait of 10 seconds. For each problem, the interviewer would record the solution children gave, whether the representation was used, and the strategy used if so. The coding of the strategy is described in the results section.
2.2.3.2 Partitioning problems
The partitioning problems were all characterised in the form of the same vignette, accompanied by an illustration (see Figure 2.2). Children were ‗introduced‘ to a character called Mary, and told that she was going shopping. Children were given an initial demonstration question to ensure understanding. They were shown a picture of three bananas and asked “can you find all the ways that Mary can put the bananas in the bags?” There were four acceptable solutions: 0 & 3, 1 & 2, 2 & 1, and 3 & 0. If children independently gave two or more solutions then the interviewer would move to the three assessed partitioning problems: “well done, see there are different ways Mary can put the fruit in the two bags”.
If children did not identify any solutions, the interviewer would support the child‘s understanding by pointing to one image of the bananas and then to one bag saying: “so, one way is there could be one banana is this bag and …” Children would then identify the second part, with the interviewer pointing to the image of the other two bags if necessary.
The interviewer would then ask children “can you find another way Mary could put the bananas in the bags?” Again, if children were not able, the interviewer would provide prompts as before for the solution 2 & 1. After children had identified at least two solutions, the interviewer would move onto the three assessed partitioning problems.
123 Figure 2.2: Supporting illustration
The children were given three partitioning problems; requiring them to partition 5, then 8, and then 10. These were the amounts used by Jones et al (1996) from which this partitioning problem was adapted. For each problem, an image of a new character was presented along with supporting images of objects to partition which were removed after asking the question. Children were introduced to the character and problem context (e.g., character packing shirts to go on holiday with two suitcases) and then asked the partitioning question: ―How many can be in each bag/suitcase?‖ Children were then reminded about using the materials for that condition: ―remember to use the cubes/squares/your fingers if they help‖.
Children were given a prompt if they did not provide a solution: “there are 5/8/10 bananas/shirts/ties, try to tell me how many can be in each suitcase/bag.” For pauses after children had identified a first solution, the interviewer would prompt: “is that all the ways or can you think of any more ways?” The session ended after two prompts had been given, or if the child said he/she had finished. If a child used non specific words such as „some‟ or „the rest‟
when identifying solutions, the interviewer would prompt by asking “so how many is
„some‟/„the rest‟?”
The interviewer recorded the verbal solutions children gave and whether the representation was used on that problem. Much positive praise was given throughout.
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