This semester I am teaching a measurement class for graduate students who themselves will teach at K-12 schools. Of course, Bond and Fox is the text-book to help my students learn about measurement so that they may be better informed teachers!
One of the most difficult concepts for my students to master has been how the Wright item-and-person maps are built. They immediately appreciate the way in the which the maps can communicate. But they become flustered with the amount of information they have to synthesize. I had asked my students to analyze the data provided for Chapter 2 of Bond and Fox. That dataset is for an elementary math test (14 children, 12 items), and the shorthand descriptions of the items provided enough information so that my graduate students could understand the items. We discussed the maps in the book, but it was clear to me that many of the students were confused.
Wright Item-Map: The item-difficulty hierarchy of the "Caring Identity" scale in Jane F. Sumner & William P. Fisher Jr. (2008) The Moral Construct of Caring in Nursing as Communicative Action: The Theory and Practice of a Caring Science. Advances in Nursing Science, 31:4, E19-E36
Then I recalled how I learned about Wright maps in stages, so I designed "hands-on" lessons for my students.
First, I provided groups of students with very large sheets of paper (about 1 meter wide and 2 meters long). I explained that we were going to try a step-by-step approach to understanding the Wright maps in Bond and Fox. My students were instructed to draw a vertical line down the left-hand-side of a sheet of paper, and mark out an equal-interval logit scale (4, 3, 2, 1, 0, -1, -2, -3, -4) from top to bottom of the sheet.
They displayed the item-measures, Table 14, in Bond&FoxSteps on their computer screens. Using this Table, they could position each item vertically on the paper, and write down the item's text at that position. Already a map was emerging from the locations of the 12 items.
The next step was to look again at Table 14 and compute the percentage of children who correctly answered the hardest item and the easiest item, writing these percentages next to the relevant item text.
Now we all discussed the ordering and spacing of items. The large sheets were easy for them to read and write upon. The percentages next to the highest and lowest items helped them understand the broad performance of the 14 children.
Following this discussion, another sheet of paper was taped to the left of the first sheet, so that it shared the same vertical logit axis. The students plotted each child's measure using the person-measures from Table 18. I also asked them to compute the success-percentage for the highest and lowest performing children. Just as was done for the easiest and hardest items, the values in percent were noted on the Wright map next to the highest and lowest performing children.
Using this extra-large Wright map, everyone could discuss the strengths and weaknesses of the test. Where were items located? How we could fill in the gaps between the existing items with new items? Did the students perform how we would have expected?
Please spend plenty of time on this activity. As we become familiar with Wright maps we forget what a challenge they are to students new to Rasch. The extra time deepens the students' comprehension of what the measures mean, so speeding up all the learning that follows. No time is lost overall and student involvement increases considerably.
The next phase in the activity involved the use of the large-scale Wright map for the creation of the Bond-and-Fox Pathway map. It is very important to first construct the measure-only Wright map (no horizontal axis), because there is a whole lot more going on measure-and-fit Pathway map! Not only do students need to think about logits, persons, items, and item descriptors, but they must also now think about fit and precision (standard errors).
Each small group of students worked with a sheet of paper about 2 meters wide and 2 meters long. I explained that we would make a Pathway map, using our completed Wright maps as a guide. The students worked out amongst themselves how to use their "old" map to plot the new map. Most students saw that they could use the location of persons and items in the old map to locate persons and items on the new map. And they realized that they now had to add a scale (left to right) on the base of the map to show fit. I helped them think through where they would find the fit data for persons and items (other columns in Tables 14 and 18). I encouraged them to first plot persons and items as dots, ignoring the standard errors of the measures. Physically plotting vertical-scale logit-values for the persons and items, and horizontal-scale fit-values for persons and items was not hard for them to do mechanically. But it was essential to give this task plenty of time, because the students have a lot to think about with the addition of the horizontal fit axis!
Once the students completed the plots, I helped them identify the columns for item error and person error. Then they used those values to draw the precision-circles for the items, and the precision-squares for the persons, centered on the dots they had already plotted. Now they had their own Pathway maps. They understood what they meant, and where they had come from. The maps made good sense. They were no longer mysterious.
Just as Ben Wright would use his yardstick-ruler while he talked, it is helpful to physically create our measuring-instruments in class. This helps Rasch novices better understand and build maps!
William Boone, Miami University (Ohio)
boonewj - at - muohio.edu
Bond & Fox Pathway Map. Bond T.G. & Fox C.M. (2001) Applying the Rasch Model: Fundamental Measurement in the Human Sciences. Fig. 3.1, p. 22.
Teaching Students about Rasch Maps W. Boone, Rasch Measurement Transactions, 2008, 22:2 p. 1163-4
|Rasch Measurement Transactions (free, online)||Rasch Measurement research papers (free, online)||Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch||Applying the Rasch Model 3rd. Ed., Bond & Fox||Best Test Design, Wright & Stone|
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