ࡱ> SUPQRa 3jbjb,, (NN]-:7    @ @ @ 4|M|M|M8M4M$4OOOOOPPP,8R=@ PP"PPP=^  OOsR^^^P* O@ O^T d |    P^^kL T@ O Kv|MWZh0z<^<^@ 44D!x/44x/Does practice make perfect?Do you play sport, play a musical instrument, learn dancing or learn mathematics? How important is practice? Practice makes perfect is a common saying since we expect to improve skills with practice. But is this true? If it is true, to what extent can we improve? What would be a typical pattern of improvement over time if we were practising a skill? To find out we can conduct experiments to measure improvement. But how can we measure improvement? What are some difficulties with measuring improvement? Discuss subjective and objective measures of improvement such as self-report and time to complete a task. If time is to be a measurement of improvement is that relevant to the skill being improved? For example: Which is the better way to measure improvement in solving mathematics problems? Time taken to solve the problems or the number of correct answers? Problem The experiment exemplified here poses the following questions: How good are you at estimating lengths within a classroom? Do your estimation skills improve with practice? An additional question for a class to think about is: Who is the best estimator in the class? Plan As a class students can discuss the test items. Consideration could be given to the magnitude of lengths to estimate and to defining the type of length (vertical, horizontal, diagonal, skew, free-form). Discussion should also centre on the questions posed and how measures can be determined and justified for capturing notions of good, improve, and best. For the questions the class can develop an overall plan for the investigation. Plan has following phases. Pre-test students by getting them to estimate the lengths of 10 objects in the room. Students analyse their data and determine how they should improve. Over the next week students practise estimating lengths. Post-test students. Students analyse their data to determine whether their estimation skills have improved with practice. As a class students analyse class data and discuss how they will decide who is the best estimator. Emma and her group, after the class discussion, decides that their test for determining how good they are at estimating length within a classroom will have 5 objects less than 1m and 5 objects over 1m, as some people might be better at estimating smaller lengths than larger lengths. Only vertical and horizontal lengths will be used as estimating skew and free-form lengths will bring in too many variables to consider as well as too many sources of variation. At the moment Emma believes that vertical and horizontal lengths involve similar skills. But Emma believes she had read somewhere that horizontal lengths are harder to estimate than vertical lengths. She has questions she might want to explore later: Should the test be solely on vertical lengths as horizontal lengths may be harder to estimate? Should she group the data on this factor in the analysis? Another concern is the order of the questions: Should the magnitudes of the lengths be mixed up? Will the order of the questions affect responses? Should the order be randomized? Should each student get the questions in a different order? Emma decides that the magnitudes and type of length will be mixed up using a random method such as putting the names of the objects to be measured on pieces of paper into a hat and then drawing out of the hat the order for estimating the lengths. The random order will prevent bias by the experimenter and any potential sources of variation. She also decides that all her group will do the same test for organisational and time reasons. Emma and her group make some further decisions: All measurements will be in centimeters. For consistency one person will measure the actual lengths and will use the same measuring tape. Good will be measured in terms of the absolute difference between the estimated length and actual length. The smaller the difference the better the estimate. Improve will be measured by considering whether a students absolute median difference has decreased and whether the spread of absolute differences has decreased. Best estimator could be measured by finding the student with the least sum of absolute differences. But student A may be quite accurate on all measurements except one, student B may not be quite accurate on all measurements yet obtain a sum less than student A. Another way could be to determine the student with the lowest median difference but the student with the lowest median may have a large spread. If two students have the same lowest median then the one with the smallest interquartile range could be the best. Emma will decide after looking at the data. Data Pre-test Data Table. Emmas pre-test estimates of lengths of objects with actual measured lengths Object Estimate (cm) Actual measurement (cm) Absolute difference Height of door 200 205 5 Length of teachers desk 140 150 10 Height of ceiling 300 289 11 Thickness of textbook 5 3 2 Height of OHP on trolley 120 155 35 Height of chair 100 78 22 Length of whiteboard pen 18 12 6 Width of computer screen 60 40 20 Height of rubbish bin 35 40 5 Length of window 140 119 21  Analysis  Pre-test Analysis Emma draws a scatter plot of the data and draws in the line for which her estimate is the same as the actual measurement. If any of her data lie on this line then she has made a perfect estimate. Data below the line suggest she underestimated the measurement while data above the line suggest she overestimated the measurement. Emma then considers other plots to draw in order to extract more information about her estimation skills from the data.  Figure 1. Scatter plot of pre-test estimate of length of object against actual length with perfect prediction line  Figure 2. Histogram of pre-test absolute differences  Figure 3. Pre-test summary statistics When Emma looked at her table of data she noticed that she often estimated to the nearest 5 or 10cm. When she looks at Figure 1 she noticed she tended to overestimate lengths less than 150cm but underestimated lengths over 150cm. She drew a histogram of the absolute differences (Fig. 2). She notices that most of her estimates are between 5 and 22cm out. Summary statistics of the absolute differences showed her median error is 10.5cm and mean error is 13.7cm (Fig. 3). The line of best fit (Fig. 4) gives her an idea of what she will estimate for given lengths which she does by either reading values from the scatter plot or by substituting into the equation of the line. For example, if the length is about 50cm her estimate will be about 55cm, that is, she is estimating about 5cm over whereas for a length about 300cm she is estimating about 5cm under From this information Emma considered how she might improve her estimation skills for lengths. She feels that she needs to practise estimating skills for lengths in the 40cm to 160cm range. By thinking about how to improve she found from the above analysis the approximate lengths to under or over estimate. For example, for large objects she will compensate her estimation by adding about 5cm. For lengths less than 40cm she needs to practise getting more accurate measurements to within 1 or 2cm. Data Post-test Data Table. Emmas post-test estimates of lengths of objects with actual measured lengths Object Estimate (cm) Actual measurement (cm) Absolute difference Height of teacher 170 174 4 Height of teachers desk 80 71 9 Length of whiteboard 330 344 14 Length of folder 25 27 2 Height of bench 98 93 7 Height of electrical plug 120 112 8 Length of whiteboard duster 15 15 0 Width of poster 40 42 2 Height of window 130 135 5 Height of filing cabinet 100 102 2 Analysis  Post-test Analysis Emma drew a scatter plot of her estimates and the actual length for the post-test and other plots to find out how she improved. Since the second question is focused on whether her estimation skills improved she compared her pre- and post- test results by using box plots.  Figure 4. Scatter plot of post-test estimate of length of object against actual length with perfect prediction line  Figure 5. Post-test summary statistics  Figure 6. Comparison of pre- and post-test absolute differences between estimate of length of object and actual length When Emma looked at her post-test scatter plot (Fig. 4) she noticed that the points lie closer to the perfect prediction line indicating her estimation skills have improved. The summary statistics also suggest she has improved (Fig, 5), because median absolute difference has decreased 6cm from 10.5cm to 4.5cm and mean absolute difference has decreased 8.6cm from 13.7cm to 5.1cm. The box plot comparison of her pre-test and post-test absolute differences (Fig. 6) found the median size of the difference decreased and that the spread of the differences is reduced. The data suggested that she has become a better and more consistent estimator of length. The absolute differences went from a range of 2cm to 35cm, to a range of 0cm to 14cm, showing she has reduced her errors (Fig. 6). She still overestimates slightly in the 50 to 140cm range (Fig. 4). For measurements below 50cm Emma considered that she has greatly improved her estimates relative to the actual measurement. Further analysis Since Emma has read that vertical lengths are easier to estimate than horizontal lengths, she re-categorised the data into horizontal and vertical lengths then compared dot plots for the pre-test and post-test.  Figure 7. Pre-test comparison of absolute differences between horizontal and vertical lengths  Figure 8. Post-test comparison of absolute differences between horizontal and vertical lengths  Figure 9. Pre-test comparison of summary statistics for absolute differences between horizontal and vertical lengths. (Row summary combines all the data)  Figure 10. Post-test comparison of summary statistics for absolute differences between horizontal and vertical lengths Emma realised that she does not have enough data to draw conclusions. This aspect of estimation should be investigated as then Emma could learn more about her estimation skills such as whether she had improved more on estimating horizontal lengths than vertical lengths or whether vertical lengths were indeed easier to estimate than horizontal lengths. Conclusion These data suggest that after one week of practice Emma has improved her estimation skills for the vertical and horizontal lengths of objects between 3cm and 344cm in a classroom. The median size of her estimation error decreased from 10.5cm to 4.5cm and the spread of her estimation errors decreased indicating she had become a more consistent estimator. ReflectionHow could this experiment be changed to investigate whether Emmas estimation skills are different for vertical and horizontal lenghts? Extension activityThere are other variations of this activity where students can do similar investigations and analyses. In particular, activities which are suited to paried comparison analyses, where students as a class collect data on themselves and compare non-independent groups. For example, compare time students can stand on one leg with eyes open to with eyes shut, compare number of whole words in the phrase the quick brown fox jumps over the lazy dog students can write in one minute with their right hand and with their left hand. Other ideas: Look at the actual difference rather than the absolute difference. Accumulate the class results, do before and after guesses for all the class.     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