Reducción de costes de transacción en la cooperación

The number of correct responses to items were analysed for distribution and central tendency [M = 32 (SD = 13.25, N = 64)]. On average, items were answered correctly by half of the participants. The range of 39 across the number of correct responses to nine items demonstrated that possessing the content knowledge to solve one problem did not necessitate being able to solve other problems related to the same content. Most teachers (N=55) correctly answered Item 2, while fewer teachers (N=16) correctly answered Item 8 than any other item.

Figure 4-4 shows the ratio of correct to incorrect responses for Problem Solving items. The number of correct responses to Item 9 was compared to the numbers of correct responses for other items to evaluate its relative level of difficulty as the item that was the basis for student work samples analysed by teachers in the Noticing Task. The number of correct responses to Item 9 (N=27) was similar to the mean number correct responses for all items, suggesting that it was within the mid-range of difficulty for teachers in the study.

Figure 4-4 Proportion of Correct: Incorrect Responses by Item

To further understand the extent to which teachers’ understandings of area, perimeter and volume supported them in solving problems, the numbers of correct and incorrect responses were examined within each of the three content ideas that were the focus of the Problem Solving Task. Teachers’ responses to questions related to the area of triangles, the relationship between area and perimeter of rectangles and the volume rectangular prisms were compared. The results, illustrated in Figure 4-

5, show that the mean numbers of correct responses (M = 31.6), (M = 30.0) and (M = 34.3) respectively, were similar across the three content ideas. On average, there was little variation in the extent to which teachers’ understandings of different ideas in the content supported them in solving problems.

Figure 4-5 Comparison of Problem Solving in Content Ideas

By comparison, the number of correct responses within content ideas did vary. The range in the number of correct responses for items assessing the relationship between the area and perimeter of rectangles (N=39) was equal to the range across all items, regardless of content (N=39). A comparison of Item 2 and Item 8, which demonstrated the greatest (N = 55) and least (N = 16) numbers of correct responses, is presented in Figure 4-6.

Greatest Number of Correct Responses

A square has an area of 121 square centimetres.

What is its perimeter?

cm

Least Number of Correct Responses

This diagram shows a common tiling pattern that uses squares and octagons.

The dotted lines show that the area of each octagon is a multiple of the area of each square.

The area of each octagon is 175 cm². What is the side length of each square?

Figure 4-6 Items Demonstrating Greatest and Least Numbers of Correct Responses

Item 8: 25% correct

Figure 4-7 illustrates that while the mean number of correct responses for Problem Solving items related to Triangles, Rectangles and Prisms were similar, the numbers of correct responses to items within the each content focus varied. In this figure, colour is used to indicate items with the same content focus.

Figure 4-7 Variation in Correct Responses by Content Focus

Table 4-2 conveys the way in which the differences in the number of correct responses, were associated with different levels of complexity in items within each content focus.

Table 4-2 Number of Correct Responses by Content Idea

Content Idea Total Responses Correct Responses Correct Responses Correct Responses Mean Range Low Complexity Moderate Complexity High Complexity Area of Triangles 192 46 27 22 31.6 24 Area of Rectangles 192 55 19 16 30.0 39 Volume of Prisms 192 43 31 29 34.3 14 Mean _{192 48 25.7 22.3 32 25.7} Range _{0 12 12 13 4.3 25}

Together, Figure 4-7 and Table 4-2 show that differences in the number of correct responses to items in the Problem Solving Task were more closely related to the complexity of items than to the content being applied. To further investigate this observation, the mean numbers of correct responses were analysed according to teachers’ ratings for familiarity, routineness and complexity. Figure 4-8 illustrates the proportion of correct and incorrect responses to items according to participants’ familiarity and complexity ratings.

Participants were almost twice as likely to correctly answer items rated as familiar (M=51) in comparison to items rated as unfamiliar (M=27). While more correct responses were observed on items rated as routine (M=38) than non-routine (M=27), routineness did not influence performance to the same extent as familiarity. When examined using teachers’ complexity ratings, complexity was identified as a factor influencing the number of correct responses to items in the Problem Solving Task. The number of incorrect responses to problems rated as low in complexity (N=16) more than doubled for problems rated as moderate in complexity (N=38). The increase in the number of incorrect responses, from items rated as moderate in complexity (N=38) to items rated high in complexity (N=42,) was less pronounced.

Analysis of the number of correct and incorrect responses to items with different characteristics showed that most teachers’ understandings of area, perimeter and volume content were sufficient to support them in solving problems that were either familiar or low in complexity, or both. However, most teachers in the study did not possess understandings of content sufficient to support them in solving problems rated as unfamiliar and/or moderate to high in complexity.

The impact of complexity on the number of correct responses was further examined by testing relationships between the number of correct responses and the number of participants rating each item as low, moderate or high in complexity. The number of teachers rating items as low in complexity was positively correlated with the number of correct responses [r = 0.923, n = 9, p = 0.01]. Conversely, the number of teachers rating items as high in complexity was negatively correlated with the number of correct responses [r = -0.767, n = 9, p = 0.05]. As the number of teachers rating an item as high in complexity decreased, the number of incorrect responses increased. Figure 4-9 captures the way that different ideas in the content, represented by colour, were distributed across complexity ratings and varied in the number of correct responses.

Figure 4-9 Correlation between Number Correct and Low - High Complexity Ratings

In summary, teachers’ understandings of content supported them in solving approximately half of the items in the Problem Solving Task. Most teachers were able to apply content knowledge to solve area, perimeter and volume problems that they rated as familiar, routine and/or low in complexity. Only one of the nine teachers with responses in the Low Problem Solving category correctly answered an item other than Item 2, which was the only item rated as familiar and routine by every participant. Most teachers’ understandings of content did not support them in solving problems rated as unfamiliar, non-routine and/or moderate-high in complexity. The item rated as unfamiliar and non- routine by all teachers, and high in complexity by most teachers, was the item with the lowest number of correct responses.