3.4.- LA COOPERACIÓN DESDE LA TEORÍA DE RECURSOS Y CAPACIDADES CAPACIDADES

Teachers’ written feedback comments on student work samples were used to confirm their rankings of work samples in the Noticing Task. While teachers provided feedback for all work samples, only feedback recorded for Amelia is used in these results. This section presents samples of feedback responses from teachers in different Noticing Task Position categories. Selected examples provide insight into what teachers in different categories attended to in Amelia’s work sample, how they interpreted her mathematical thinking, and how they responded to her the understanding and reasoning she had demonstrated. The examples presented summarise the range of feedback in teachers’ responses to the same student work sample. An image of a ranking and written feedback comment on Amelia’s work sample is presented in Figure 4-12 to illustrate the connection between the work sample, the teacher’s ranking and feedback in the Noticing Task.

Figure 4-12 Sample Ranking and Teacher Feedback Comment

Teachers who interpreted Amelia’s work sample as Limited or Basic tended to misinterpret her strategy of rearranging the areas of the triangles to form a rectangle by using the common side length and averaging the perpendicular heights to establish the width of the rectangle. It is important to recall that none of the teachers in these two categories had correctly responded to this item in the Problem Solving Task. Most responses ranking Amelia’s work as Limited or Basic interpreted her explanation of rearranging area as evidence that she thought that the shape was a rectangle or did not know how to calculate the area of a triangle. None of the responses in these Noticing Task positions provided feedback that was likely to enhance Amelia’s reasoning or understanding for calculating area. The range of responses in the two lowest Noticing Task Positions included:

 Comments framed as questions, e.g. “Can you explain where you got the 18 m from? This information is not in the question”.

 Explanations that the shape was not a rectangle and comprised three, rather than two, triangles, e.g. “This is not a rectangle. You need to check whether the sides are parallel. The shape is actually made up of three triangles, not two. You need to calculate the area of each of the three triangles first and then add the three of them together”.

 Suggestions that Amelia had omitted a step, e.g. “Explain what you need to do after the rectangle. There is another step. You need to halve your answer at the end and use the formula for the area of a triangle”.

Some of the teachers who ranked Amelia’s work sample in the Sound position provided feedback comments recognising that her answer was correct. However, most teachers in this category interpreted Amelia’s efficient solution as an absence of procedural knowledge and therefore a deficit in her thinking. In considering these feedback responses, it is important to remember that the majority of teachers with responses in this category had not solved this item in the Problem Solving Task. Generally, feedback from teachers in this position would be unlikely to enhance Amelia’s understandings of area measurement. The range of feedback from teachers with responses in this position included:

 Comments conveying that it was not possible to work out the answer using Amelia’s strategy, e.g. “You will not get the correct answer using this strategy. Try making your calculations in smaller steps to avoid making unnecessary mistakes”.

 Suggestions that Amelia’s method only worked by coincidence and that a formula would increase accuracy, e.g. “This is a great attempt at working out a problem using your own approach. You managed to get the correct answer but now you need to learn the correct formula for calculating the area of a triangle and set out your work using the formula so that it is more accurate”.

 Recognition of Amelia’s approach while also seeking a more detailed explanation of how and why the approach worked, e.g. “This looks like a clever idea. Check your working using a different strategy to see if you get the same answer using a different method. I would like to hear more about how this strategy works as I am not certain about whether it would work on other triangle questions”.

Most teachers who ranked Amelia’s work sample as Thorough or Extensive tended to be confident that her answer was correct. Feedback from teachers in these categories generally acknowledged the

elegance of Amelia’s strategy with some suggestions about how to improve the mathematical communication in the work sample. Comments included:

 Admitting that the strategy worked without recognising why or how it might be extended, e.g. “You managed to achieve the correct answer this time; however this strategy will not work for other examples. To improve your work please use the correct formula”.

 Stating that the answer was correct and conveying a preference for the use of a formula, e.g. “Your calculation is correct. Next time please show all of your working and also show that you can check you work using the appropriate formula”.

 Recognising the correctness of the answer and suggesting ways to improve the mathematical communication, e.g. “Your answer is correct, but you need to make your communication clearer so that the teacher can understand what you are doing. Try shading the diagram to show where the pieces of the triangles move to form the rectangle and why this works”.  Acknowledging the elegance of the solution and proposing that Amelia further develop her

theory, e.g. “This is an incredibly efficient way of working out the area of this shape. It would be good to investigate how and when you could apply this thinking to a range of other shapes. I would like you to concentrate on providing more detail about why you could average the heights of the triangles”.

The latter feedback comment on a response in the Extensive Noticing Task Position Category provides an example of feedback that could provide some direction for Amelia on how to improve her response to the problem.

In summary, most teachers did not interpret a work sample with a correct, efficient, sophisticated solution, as representing Extensive achievement. Teachers who had successfully solved the problem on which the work samples were based were far more likely to rank Extensive mathematical thinking in the Thorough or Extensive positions. Only four teachers who had not solved the problem correctly themselves identified Amelia’s work sample as Extensive or Thorough. Written feedback responses from teachers in each Noticing Task Position Category supported the analysis of numerical data. Generally, written feedback suggested that most teachers attended to the absence of a formula, rather than to the reasoning and understanding of area relationships presented by Amelia. Written feedback also illuminated that teachers with incorrect solutions to this problem were uncertain about whether Amelia’s solution was correct. This contributed to the ranking of Amelia’s work sample as Basic or Limited. The section that follows presents Noticing Task Student Category results to capture which student work sample teachers ranked in the Extensive position.