Knowledge and Judgment Scoring - Operational to Instructional

                                                              23

This post (and the next three) introduce why we need a KJS Group. The software, Power Up Plus (PUP), that contains both Knowledge and Judgment Scoring (KJS) and traditional multiple-choice (TMC) is now free to registered KJS Group members.  Version 5.22, is free to teachers and administrators. Please see instructions below. 

This reflects a change in use of the software as an operational program for scoring individual classroom tests, to use as an instructional program to promote student and teacher development in preparation for the CCSS movement assessments. Students and teachers can readily see the difference between lower and higher levels of thinking when students are offered the opportunity to report, in a non-threatening environment, what they actually trust they know and can do, that serves as the basis for further learning and instruction. Practice riding the tricycle is poor preparation for a riding test on a bicycle.

Last week I finished a series of 22 posts on this Multiple-Choice Reborn blog. The series makes clear, that no amount of “statistical work” can extract from TMC marked answer sheets, some of the claims now being marketed about them. These tests can, at best, only do a good job of ranking students.

They so imperfectly and incompletely tell us what students know and can do that North Carolina is now spending six months figuring out how and where to place the cut scores on their new CCSS traditionally scored end-of-grade, multiple-choice math test results. 

[They must guess where to put the cut score on the results from uncommitted, low scoring, improperly prepared students, who were guessing at the right answers to questions the test maker guessed, would produce a satisfactory score distribution, with high statistical reliability and precision. The more nonsensical the student mark data are, the more subjective the process.]

Accurate, honest, and fair testing can be done with Knowledge and Judgment Scoring and the partial credit Rasch model analysis. These methods allow students to report what they actually know and can do that is meaningful, useful, and empowering. Student development (the judgment to appropriately use all levels of thinking) is as important as knowledge and skills for successful students and employees (Knowledge Factor). 

The NCLB decade has laid the foundation for real change by making schools designed for failure (that promote students beyond their abilities, rather than developing the necessary abilities for their success) so bad and so visible, that something had to be done. The CCSS movement has rekindled the old alternative (to TMC) testing and authentic testing methods; with the addition of CAT and elaborate assessment methods.

My concern now is that, after expending a large amount of time and money on promoting the CCSS movement ideals, a major part of the assessments will once again be reduced back again to traditional guess testing at the lowest levels of thinking. 

Both KJS and TMC scoring can use the same test questions. In fact both methods are used on the same test to accommodate students working at all levels of thinking and with all degrees of preparation (PUP).

IMHO, KJS is a practical method of achieving the CCSS movement goals. It prepares students for  standardized tests presented at all levels of thinking.  [I still cannot predict when KJS or the partial credit Rash model will be used on standardized tests as current standardized tests are not designed to assess what students know or can do. They are designed, using the fewest questions, to produce an acceptable spread of student scores.]

Rather than a rank of 60 on a test, a student may get a quality score of 90% on questions used to report what the student actually knows and can do, as well as, a rank of right marks on the test using KJS. We now know what a “just passing” student knows with the same accuracy as a student earning a 90% score on a traditional test. This can be valuable formative assessment information. 

Letting students tell us what they know or can do makes more sense than the guessing game now in use during preparation and assessment. And over 90% of my students preferred Knowledge and Judgment Scoring after just two experiences with it. Even students like an honest and fair test over gambling for a grade.

Past performance in my classroom is no guarantee of performance in your classroom unless you are a likeminded teacher, administrator, or test maker.

[The Educational Software Cooperative, Inc. (non-profit) closed this year (2013) after 20 years of operation during which I was the volunteer treasurer. It was founded to maximize the benefits of an individual computer: infinite patience, non-judgmental, and best of all, instant formative feedback. That level of instruction and record keeping has now been surpassed by the necessity for district wide record keeping systems operating online assessments keyed to CCSS learning objectives.]

Welcome to the KJS Group: Please register at mailto:KJSgroup@nine-patch.com. Include something about yourself and your interest in student empowerment (your name, school, classroom environment, LinkedIn, Facebook, email, phone, and etc.).

Free anonymous download, Power Up Plus (PUP), version 5.22 containing both TMC and KJS: PUP522xlsm.zip, 606 KB or PUP522xls.zip, 1,099 KB.

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Free software to help you and your students experience and understand how to break out of traditional-multiple choice (TMC) and into Knowledge and Judgment Scoring (KJS) (tricycle to bicycle):

• Break Out Overview
• Download Break Out
• Quick Start

Multiple-Choice Test Analysis - Summary

                                                              22

The past 21 posts have explored how classroom and standardized tests are traditionally analyzed. The six most commonly used statistics are made fully transparent in Post 10, Table 15, the Visual Education Statistics Engine (VESE) [Free VESEngine.xlsm or VESEngine.xls]. One more statistic was added for current standardized tests. Numbers must be meaningful, understood; to have valid, practical value.

•       Count: The count is so obvious that it should not be a problem. But it is a problem in education.  Counting right marks is not the same as counting what a student knows or can do. Also a cut score is often set by selecting a point in a range from 0% to 100%. A cut score of 50 means 50%. But the test, when administered as traditional multiple-choice starts each student at 25% with 4-option questions. [There is no way to know what low scoring students know, only their rank.]
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•       Average: Add up all of the individual student scores and divide by the number of students for the class or test average score. [There is no average student.] Classes or tests can be compared by their averages just as students can be compared by their counts or scores.
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•         Standard Deviation (SD): Theoretically, 2/3 of the counts on a distribution of scores are expected to fall within one SD of the average. A very well prepared (or very under prepared) class will yield a small SD. A mixed class will yield a large SD with students with both very high and very low scores (many A-B and D-F, with few C grades).
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•       Item Discrimination: A discriminating question groups those who know (high scoring students) into one group and those who do not know (low scoring students) into another group. Every classroom test needs about ten of these to produce a grade distribution where one SD is ten percentage points (a ten point range for each grade).
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•       Test Reliability: A test has high reliability when the results are highly reproducible. Standardized tests, therefore, use only discriminating questions. They rarely ask a question that almost all students can answer correctly. Traditional multiple-choice, therefore, does not assess what students actually know and value. Traditional standardized tests can only rank students.
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•       Standard Error of Measurement (SEM): Theoretically, 2/3 of the time a student retakes the same test; the scores are expected to fall within one SEM of the average. The SEM value fits inside the range of the SD. “Jimmy, you failed the test, but based on your test score and your luck on test day, each time you retake the test, you have a 20% expectation of passing without doing any more studying.” The SEM precision is based on the reliability of the entire test.
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•       Conditional Standard Error of Measurement (CSEM): The CSEM is based (conditioned) on each test score. This refinement in precision is a recent addition to traditional multiple-choice analysis. It has been a part of the Rasch model IRT analysis for decades.

Even the CSEM cannot clean up the damage done by forcing students to mark every question even when they cannot read or do not understand the question. Knowledge and Judgment Scoring and the partial credit Rasch model do not have this flaw. Both accommodate students functioning at all levels of thinking and all levels of preparation.  These two scoring methods are in tune with the objectives of the CCSS movement.

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Free software to help you and your students experience and understand how to break out of traditional-multiple choice (TMC) and into Knowledge and Judgment Scoring (KJS) (tricycle to bicycle):

• Break Out Overview
• Download Break Out
• Quick Start

Visual Education Statistics - Conditional Standard Error of Measurement

                                                                 21

[[Second Pass, 8 July 2014.  Equation 6.3 (cited below) in Statistical Test Theory for the Behavioral Sciences by Dato N.M. de Gruijter and Leo J. Th. van der Kamp, 2008, is the same as the calculation used in Table 29, in my 9 July 2014 post. On the following page they mention that the error variance is higher in the center and lower at the extremes. That distribution is the green curve on Chart 73. I did not see this relationship in the equation when this post was first posted, but do now in the visualized mathematical model (Chart 73).

Also the discussion of Table 24 has been updated to match the terms and values in Table 24.]]

Working on the conditional standard error of measurement (CSEM) is new territory for me. I always associated the CSEM with the Rasch model IRT analysis commonly used by state departments of education when scoring NCLB tests. I first had to Google for basic information.

If you are interested in the details, please check out these sources for sample (n-1) equations: (Equation 6.14 that corrects the relative variance was not included in the 2005 version of the current 2008 version. This represents a significant progress in applying test precision.)

•        Absolute Error Variance                 Equation 5.39 p. 73
•        Relative Error Variance                  Equation 6.3 p. 83
•        Corrected Relative Variance           Equation 6.14 p. 91 or GED Equation 3 p. 9

My first surprise was to find I had already calculated the CSEM for the Nursing124 data when I put up Post 5 of this series (in Table 8. Interactions with Columns [Items] Variance, MEAN SS = 3.33) as I discovered five ways to harvest the variance [mean sum of squares (MSS)]. Equation 6.3 n, Table 22, produces the same result (test SEM = 1.75) when it divides by n [unknown population] rather than n-1 [observed sample].

[n = the item count. Test SEM = AVERAGE(CSEM).]

I then used what I learned in the last post to table data to obtain the conditional error variance for student scores (Table 23a). The 21 items in Table 22 became the number of right marks on each of 11 item difficulties on Table 23a. The values in this tabulation were then converted into frequencies conditional on the student scores; the sum of which added to one, for each score (Table 23b).

The absolute error variance for each score was computed by Excel (=Var.P). Multiplying the absolute error variance (0.14382) by the square of the item count (21^2) yields the relative error variance (63.42). [Equation 5.39 (0.14382) * n^2 = Equation 6.3 (63.42)] The square root of the relative error variance of each score yields the CSEM for that score. [An alternate calculation of the absolute error variance is shaded in Table 23b. Here the variance was calculated first and that value divided by the squared score to obtain the absolute error variance. This helps explain multiplying the absolute error variance by the squared item count to obtain the relative error variance for each score.]

The conditional frequency estimated test SEM was 1.68 (Table 23b). The conditional frequency CSEM values for each score were different for students with the same score. The CSEM values had to be averaged to get results comparable with the other analyses. These values generated an irregular curve, unlike the smooth curve for the other analyses (Chart 61). The conditional frequency CSEM analysis is sensitive to the number of items with the same difficult (yellow bars alternate for each change in value, Table 23b). The other analyses are not sensitive to item difficulty (yellow bars, in Table 22, include all students with the same score).

Complete curves were generated from Equation 6.3 for n-1 and for GED n-1 (Table 24). The GED n-1 analysis includes a correction factor (cf) for the range of item difficulties on the test [cf = (1- KR20)/(1-KR21)]. This factor is equal to one if all items are of equal difficulty. For the Nursing123 data it was 1.59; the difficulties ranged from 45% to 95%, from the middle of the total possible distribution to one extreme.

The CSEM values from the six analyses are listed in Table 24. Five are fairly close to one another. The GED n-1, with a correction for the range of item difficulties, is far different from the other five (Chart 61). Values could not be created for the full curve for conditional frequencies as you must actually have student marks to calculate conditional frequency CSEM values. The gray area shows the values calculated from an equation for which there were no actual data. Equations produce nice looking, “look right”, reports.

The CSEM improves the reportable precision on this test over using the test SEM. Good judgment (best practice) is to correct the CSEM values as done on the GED n-1 analysis.

[I did not transform the raw test score mean of 16.8 or 79.8% to a scale score of 50% as was done by Setzer, 2009, GED, p. 6 and Tables 2 and 3. The GED n-1 raw score cut point was 60% which is comparable to most classroom tests. If 25% of the score is from luck on test day that leaves 35% for what a student marked right as something known or could be done, as a worst case. If half of the lucky marks were also something the student knew or could do, the split would be about 10% for luck on test day and 50% for student ability.]

In Table 24, the GED n-1 analysis test SEM of 2.98 for the Nursing124 data is, as a range, 2.98/21 or 14.19%. For the uncorrected Equation 6.3 n-1 analysis, 1.79, the range is 1.79/21 or 8.52%. The n SEM was 1.75 or 7.95%. The n SEM range, 1.75, fits within the uncorrected n - 1 test SEM value, 1.79. The corrected GED n-1 test SEM value, 2.98, exceeds it.

Student score CSEM values are even more sensitive than the test SEM values. The maximum range for the GED n-1 analysis is 3.73 or 3.73/21 or 17.76% and for the Equation 6.3 n-1 analysis 2.35 or 11.19%. Both are beyond the maximum n CSEM value of 2.29 or  10.41%. This low quality set of data fails to qualify as a means of setting classroom grades or a standardized test cut score.

[However the classroom rule of 75% for passing the course and the rule for grades set at 10 percentage points over rule these statistics. Here is a good example that test statistics have meaning only in relation to how they are used. If the process of data reduction and reporting is not transparent, the resulting statistics are suspect and can produce extended debates over a passing score in the classroom.]

The CSEM for each student score does improve test precision. It can be calculated in several ways with close agreement. But it cannot improve the quality of the student marks on the answer sheets made under traditional, forced-choice, multiple-choice rules. These tests only rank students by the number of right marks. They do not ask students, or allow students to report, what they really know or can do; their judgment in using what they know or can do.

The CCSS movement is now promoting learning at higher levels of thinking (problem solving) with, from which I have learned, some de-emphasis  on lower levels of thinking that are the foundation for higher levels of thinking. A successful student cycles through all levels of thinking, as is needed. Yet half of the CCSS testing will be at the lowest levels of thinking, traditional multiple-choice scoring. The other half will be as much of an over kill as traditional multiple-choice is an under kill in assessing student knowledge, skills, and student development to learn and apply their abilities. Others have this same concern that centralized politics (and dollars) will continue to overshadow the reality of the classroom.

There is a middle ground that makes every question function at higher levels of thinking, allows students to report what is meaningful, of value, and empowering, and has the speed, low cost, and precision of traditional multiple-choice. Knowledge and Judgment Scoring and partial credit Rasch model IRT are two examples. They both accommodate students functioning at all levels of thinking. Lower ability students do not have to guess their way through a test. With routine use, both can turn passive pupils into self-correcting highly successful achievers in the classroom. If you are really into mastery learning, you can also try something like Knowledge Factor.

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Free software to help you and your students experience and understand how to break out of traditional-multiple choice (TMC) and into Knowledge and Judgment Scoring (KJS) (tricycle to bicycle):

• Break Out Overview
• Download Break Out
• Quick Start

Visual Education Statistics - Frequency Estimation Equating

                                                                20

Frequency Estimation Equating involves conditioning on the anchor or a set of common items. This post reports my adventures in figuring out how this is done as I needed to know how to do this to complete the next post on conditional standard error of measurement (CSEM).

Two 24 student by 15 item tests were drawn from the Nursing124 data. Included in each was a set of 6 common items that were marked the same in both tests: A and B (Table 20). Student scores varied between Test A and Test B based on their marks on other, non-common, items. The common items were sorted by their difficulty.

I then followed the instructions in Livingston, (2004, pp. 49-51). The values in Table 20 were tabulated to produce “a row for each possible [student] score” and “a column for each possible score on the anchor [common item]” (Table 21). The tally is turned into frequencies conditioned on the common item scores by dividing each column cell by the common item score or difficulty. The frequencies for each common item sum to 1.00.

Next, the unknown population proportions are obtained by combining (multiplying) the common item frequencies with the equal portion each common item contributed (1/6) to the test (Table 21). These values now represent the on-average expectations for each cell based on the observed data. Summing by rows produces the estimated (best guess) unknown population student score distribution that could have also produced the on-average expectations. This was done for both Test A and Test B.

[This operation can be worked backward (in part) to yield the right mark tally. Dividing the population proportions by the number of items in the sample yields the right mark frequencies. Multiplying the right mark frequencies by the difficulty yields the right mark tally. But there is no way to back up from the estimated population distribution to this set of population proportions, let alone to individual student marks. The right mark tally is a property of the observed sample and of individual student marks. This estimated population distribution is a property of the unknowable population distribution related to the normal curve. The unknowable population distribution can spawn endless sets of population proportions. Monte Carlo psychometric experiments can be clean of the many factors that affect classroom and standardized test results.]

Charts 59 and 60 show the effect produced by conditioning on the common items. This transformation from observed to on-average expectations appears to rotate the distribution about the average test score of 84% and 80%, respectively, for both Test A and Test B. It made a detectable increase in the frequency of high scores and a similar decrease in the frequency of low scores. This increased the average scores to 86% and 84%, respectively. Is this an improvement or a distortion?

“And when we have estimated the score distributions on both the new form and the reference form, we can use those estimated distributions to do an equipercentile equating, as if we had actually observed the score distributions in the target population.” I carried this out, as in the previous post, with nothing of importance to report.

So far in this series I have found that data reduction from student marks to a finished product is independent from the content actually on the test. The practice of using several methods and then picking the one that “looks right” has been promoted. Here the creation of an unknown population distribution is created from observed sample results. Here we are also giving the choice of selecting Test A or Test B or combining the results. As the years pass, it appears that more subjectivity is tolerated in getting test results that “look right” when using traditional, non-IRT, multiple-choice scoring. This charge, formerly, was directed at the Rasch model IRT analysis.

It does not have to be that way. Knowledge and Judgment Scoring and partial credit Rash model IRT allow a student to report what is actually meaningful, useful, and empowering to learn and apply what has been learned. This property of multiple-choice is little appreciated.

What the traditional multiple-choice is delivering is also little understood (psychometricians guessing to what extent sample [actual test] results match an unknowable standard distribution population based on student marks that include forced student guessing on test items the test creators are guessing students will find equally difficult, as based on a field test, they guess will represent the current test takers, on average).

We still see people writing, “I thought this test was to tell us what [individual] students know.” Yet, traditional, forced-choice, multiple-choice can only rank students by their performance on the test. It does not ask them, or permit them, to individually report what they actually know or can do based on their own self-judgment: just mark every item (as a missing mark is still considered more degrading to an assessment than failing to assess student judgment).

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Free software to help you and your students experience and understand how to break out of traditional-multiple choice (TMC) and into Knowledge and Judgment Scoring (KJS) (tricycle to bicycle):

• Break Out Overview
• Download Break Out
• Quick Start

Visual Education Statistics - Equipercentile Equating

                                                             19

Equipercentile equating frequently appears in NCLB testing articles. I took a normal distribution of 40 student scores (average of 50%) with a standard deviation (SD) of 10% (new test) and equated it to one with a SD of 20% (reference test) to see how equipercentile equating works (Chart 54).

First I grouped the scores into 5%-ranges. I then matched the new test groups to the reference test groups (Chart 55). The result was a bit messy. 

A re-plot of the twenty 5%-groups shows the new test has been sliced into groups that contain twice the count as the reference test, but which match, in general, the reference test every other group (Chart 56).

Smoothing by inspection resulted in Chart 57. A perfect fit was obtained with the reference test with the exception of rounding errors. 

Smoothing on “small samples of test-takers” does make a difference in the accuracy of equipercentile equating. “The improvement that resulted from smoothing the distributions before equating was about the same as the improvement that resulted from doubling the number of test-takers in the samples” (Livingston, 2004, page 21). [See Post 13, Chart 34, in this series for the effect of doubling the number of test-takers on the SD and SEM.]

I then entered the values from Charts 54, 55, and 57 into my visual education statistics engine (VESE). Equipercentile equating the student scores transformed the new test into the reference test including the related group statistics (Chart 58).

The three 5%-groupings show almost identical values. Grouping reduced the item discrimination ability (PBR) of the reference test a small amount as grouping reduced the range of the student score distribution. This works very nicely in a perfect world, however, real test scores do not align perfectly with the normal curve.

A much more detailed description of equipercentile equating and smoothing is found in (Livingston, 2004, pages 17-24). The easy to follow illustrated examples include real test results and related problems, with a troubling resolution: “Often the choice of an equating method comes down to a question of what is believable, given what we know about the test and the population of test-takers.”

This highly subjective statement was acceptable in 2004. NCLB put pressure on psychometricians to do better. The CCSS movement has raised the bar again. The subjectivity expressed here is, IMHO, similar to that in using the Rasch model IRT analysis that has been popular with state departments of education. Both without IRT and with IRT methods base results on a relationship to an unknowable “population of test-takers”. Both methods pursue manipulations that end up with the results “looking right”.

[The classroom equivalent of this, practiced in Missouri prior to NCLB, was to divide the normal curve into parts for letter grades. One version was to assign grades to ranked student scores with uniform slices. True believers assigned a double portion to “C”. Every class was then a “normal” class with no way to know what the raw scores were or what students actually knew or could do.]  

It does not have to be that way. Let students report what they actually know and can do. Let them report what they trust will be of value for further learning and for application in situations other than in which they learned. Do multiple-choice right. Get results comparable to essay, project, report, and research. Promote student development. Knowledge and Judgment Scoring and partial credit Rasch model analysis do this. Guessing is no longer needed. Forced guessing should not be tolerated, IMHO.

The move to performance based learning may, this time, not only compete with the CCSS movement assessments, but replace them. The system that is the leanest, the most versatile in meeting student needs, and is immune to erratic federal funding, and thus most effective, will survive.

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Free software to help you and your students experience and understand how to break out of traditional-multiple choice (TMC) and into Knowledge and Judgment Scoring (KJS) (tricycle to bicycle):

• Break Out Overview
• Download Break Out
• Quick Start

Visual Education Statistics - Equating

                                                             18

The past few posts have shown that if two tests have the same student score standard deviation (SD) they are easy to combine or link. Both tests will have the same student score distribution on the same scale.

Equating is then a process of finding the difference between the average test scores and applying this value to one of the two sets of test scores. Add the difference in average test score to the lower set of scores, or subtract it from the higher set to combine the two sets of test scores.

This can be done whenever the SDs are within acceptable limits (considering, all factors that may affect the test results, the expected results, and the intended use of the results). This is IMHO a very subjective judgment call to be made by the most experienced person available.

There are two other situations: same average test score but the different SDs are beyond acceptable limits, and both test score and SD differences are beyond acceptable limits for the two tests. In both cases we need to equate the two different SDs, the two different distributions of student scores.

Chart 48 is a re-tabling of Chart 44. The x-axis in Chart 48 shows the set Standard Deviation (SD) used in the VESE tables in prior posts. Equating a low SD test (10) to a high SD test (30) has different effects then equating a high SD test (30) to a low SD test (10). The first improves the test performance; the second reduces the test performance.

There is then a bias to raise the low SD test to the high SD test. “The test this year was more difficult than the test last year,” was the NCLB explanation from Texas, Arkansas, and New York. [It was not that the students this year were less prepared.]

The most frequent way I have seen mapping (Livingston, 2004, figure 2, page 14) done is to plot the scores of the test to be equated on the x-axis and the scores of the reference test on the y-axis. The equate line for two tests with similar average test scores and SDs is a straight line from zero through the 50% point on both axes (Chart 49).

If the average test scores are similar but the SDs are different, the equate line becomes tilted to expand (Chart 50) or contract (Chart 51) the equated values to match the reference test. Mapping from a low SD test to a higher SD tests leaves gaps. Mapping from a high SD test to a low SD tests produces clumping, in part, from rounding errors.

Mapping a new difficult test to an easier reference test with the same SD increases the values on the equating line, as well, as truncates it. Any new test scores over 30 on Chart 52 have no place to be plotted of the reference test scale. 

The equating with an increase in both SD and average test score expands the distribution and truncates the equating line even more (Chart 52). A comparison of the two above situations as parallel lines (Chart 53) helps to clarify the differences.

Both increase the new difficult test average test score value of 20 counts to 30 counts on the reference scale. In this simple example based on a normal distribution, the remaining values increase in a uniform manner of equal units of 10 with the same SD and 15 when mapping to the larger SD.

The significance of this is that in the real world, test scores are not distributed in nice ideal normal distributions. The equating line can assume many shapes and slopes.

The unit of measure needed to plot an equating chart must include equivalent portions of the two distributions. Percentage is a convenient unit: equipercentile equating. [More on this in the next post.]

Whither Test A is the reference test, or Test B is the reference test, or both are combined as one analysis is the difficult subjective call of the psychometrician. So much depends on the luck on test day related to the test blueprint, the item writers, the reviewers, the field test results, the test maker, the test takers and many minor effects on each of these categories. 

This is little different from predicting the weather or the stock market, IMHO. [The highest final test scores at the Annapolis Naval Academy were during a storm with very high negative air ion concentrations.] The above factors also need to include the long list of excuses built into institutionalized education at all levels.

On a four-option item, chance alone injects an average 25% value (that can easily range from 15 to 35%) when students are forced to mark every item on a traditional multiple-choice (TMC) test. Quality is suppressed into quantity by only counting right marks: Quality and quantity are therefore linked into the same value. TMC high test scores have higher quality then lower test scores, but this is generally ignored.

It does not have to be that way. Both the partial credit Rasch model IRT and Knowledge and Judgment Scoring permit students to report what they trust they know and can do and what they have yet to learn accurately, honestly and fairly. No guessing is required. Both paper tests and CAT tests can accept, “I trust I know or can do this,” “I have yet to learn this,” and if good judgment does not prevail, “Sorry, I goofed.”  Just score 2, 1, and 0 rather than 1 for each right mark (for whatever reason or accident).

A test should encourage learning. The TMC at the lower scores is punitive. By scoring for both quantity and quality (knowledge and judgment) students receive separate scores, just as is done on most other assessments. “You did very well on what you reported (90% right) but you need to do more to keep up with the class” rather than “You failed again with a TMC score of 50%.

Classroom practice during the NCLB era tragically followed the style of the TMC standardized tests conducted at the lowest levels of thinking. The CCSS tests need to model rewarding students for their judgment as well as right marks. [We can expect the schools to again doggedly try to imitate.] It is student judgment that forms the basis for further learning at higher levels of thinking: one of the main goals of the CCSS movement. The CCSS movement needs to update its use of multiple-choice to be consistent with its goals.

Equating TMC meaninglessness does not improve the results. This crippled form of multiple-choice does not permit students to tell us what they really know and can do that is of value for further learning and instruction.

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Free software to help you and your students experience and understand how to break out of traditional-multiple choice (TMC) and into Knowledge and Judgment Scoring (KJS) (tricycle to bicycle):

• Break Out Overview
• Download Break Out
• Quick Start

Visual Education Statistics - Standardized Tests

                                                              17

Standardized test makers use statistics to predict what may happen; classroom statistics describe what has happened. Classroom tests include two or three dozen students. Standardized test making requires several hundred students. Classroom tests are given to find out what a student has yet to learn and what has been learned. Standardize tests are generally given to rank students based on a benchmark test sample. Classroom and standardized tests have other significant differences even though they may use many of the same items.

I took the two classroom charts (37 and 38 in a previous post) and extended the standard deviations (SD) from 5 - 10%, to 10 - 30%; a more realistic range for standardized tests (Chart 44). At a 70% average score and 20% SD the normal curve plots of 40 students by 40 items started going off scale. I then reversed the path back to the original average score of 50% as the SD rose from 20% to 30%.

The test reliability (KR20) continued to rise with the SD for these normal distributions set for maximum performance. The item discrimination (PBR) rose slightly. The relative SEM/SD value decreased (improved) from 0.350 to 0.157 as test reliability increased (improved).

The two tests with average test scores of 50% yielded very different test reliability and item discrimination values for SD values of 10% and 30% on Chart 44; the greater the distribution spread, the higher the KR20 and PBR values. [I plotted the N – 1 SD to show how close the visual education statistics engine (VESE) tables were to their expected normal curves.]

The SD is then a key indicator of test performance; the spread of the student score distribution, the main goal for standardized test makers. It is also very sensitive to extreme values. The 30% SD plot was made by teasing the VESE table that I set for 30% SD. The original SD value was near that for a perfect Guttman table (each student score and each item difficulty appear only once), about 28%. By moving four pair of marks, near the extreme ends of the distribution, one count more toward the end, the SD rose to 30%. That is moving four pair of marks out of 400 pair one count each to change the SD by 2%.

The standard error of measurement (SEM) under optimum normal test conditions remained about 4.4% (Chart 44). So, 4.4 x 3 = 13.2%. A difference in a student’s performance of more than 13.2% would be needed to accept the scores as representing a significant improvement with a test reliability of 0.95. All of the above mark patterns were not mixed; which is an unrealistically optimum performance.

I looked again at the effect of mixing right and wrong marks on an item mark pattern with a higher SD value than found in the classroom (Chart 45). The change from a SD of 10% to 20% was much smaller than I had anticipated. The effect of deeper mixing was again linear.

Average item difficulty sets limits on the maximum PBR that can be developed (Chart 46). In a perfect world where all items are marked either all right or all wrong, the maximum PBR is 1.0 for individual items.

Looking back at prior posts, I found lower values on a perfect Guttman table (0.84) and a normal curve table set at 30% SD (0.85). The PBR declined along with the SD set to 20% and 10% (Chart 46). 

These values hold for tests with average test scores that range from 50% to 70%.

There is now enough information to construct the playing field upon which psychometricians play (Chart 47).  I chose two scoring configurations: Perfect World and Normal Curve with a SD of 20%. The area in which standardized tests exit is a small part of the total area that describes classroom tests. The average student score and item difficulty were set at 50%.

An item mark pattern at 50% difficulty can produce a PBR of 1.0 in a perfect world (blue). All right marks are together and all wrong marks are together. The PBR drops to zero with complete mixing (Table 20). It falls to a -1.0 when all right marks are together at the lower end of the mark pattern.

The area for the normal curve distribution (red) with a SD of 20% fits inside the perfect world boundary. This entire area is available to describe classroom test items. Items that are easier or more difficult than 50% reduce the maximum possible PBR. They have shorter mark patterns. And here too, fully mixed patterns drop the PBR to zero.

We can now see the problem psychometricians face in making standardized tests. The standardized test area is about 1/8^th of the classroom area. Standardized tests never use negative items (that almost excludes misconceptions which cannot be distinguished from difficult items using traditional multiple-choice scoring; as they can using Knowledge and Judgment Scoring).

Chart 44 indicates an average PBR of over 0.5 is need for the desired test reliability of over 0.95 under optimum conditions (no mark pattern mixing). With just ¼ mixing, the window for usable items becomes very small. The effect of mixing right and wrong marks on an item mark pattern varies with item difficulty. A test averaging 75% right with unmixed items would be the same as a test averaging 50% right with partially mixed items.

A 2008 paper from Pearson, by Tony D. Thompson, confirms this situation. “This variation, we argue, likely renders non-informational any vertical scale developed from conventional (non-adaptive) tests due to lack of score precision” (page 4). “Non-informational” means not useful, not valid, does not look right, and does not work, IMHO. “Conventional” means, in general, paper tests and the fixed form tests being developed by PARCC for online delivery for the Common Core State Standards (CCSS) movement.

This comment may be valid for “many educational tests” (page 14). “Also, if an individual’s observed growth is much larger than the associated CSEM, then we may be confident that the individual did experience growth in learning.” This indicates that using simulations within the playing field, as Thompson did, confirms my exploration of the limits of the playing field. [And the CSEM, which is applied to each score, is more precise than the SEM based on the average test score.]

“While a poorly constructed vertical scale clearly cannot be expected to yield useful scores, a well-defined vertical scale in and of itself does not guarantee that reported individual scores will be precise enough to be support meaningful decision-making” (page 28). This cautionary note was written in 2008, several years into the NCLB era.

The VESE tables indicate that the “best we can do” is not good enough to satisfy marketing department hype (claims). Testing companies are delivering what politicians are willing to pay for: a ranking of students, teachers, and administrators only based on a test producing scores of questionable precision. Additional use of these test scores is problematic.

An unbelievable situation is currently being challenged in court in Florida. Student test scores were used to “evaluate” a teacher who never had the students in class! It reveals the mind set of people using standardized test scores.  They clearly do not understand what is being measured and how it is being measured. [I hope I do by the end of this series.] Just because something has been captured in a number does not mean that the number controls that something.

Scoring all the data that can be in the answer sheets would provide the information (which is repeatedly sought but ignored in traditional multiple-choice) needed to guide student, teacher and administrator development. Schools designed for failure (“Who can guess the answer?”), fail. Schools designed for success have rapid, effective, feedback with student development (judgment) held as important as knowledge and skills. Judgment comes from understanding, a goal of the CCSS movement.

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