Quantitative Evidence of Student Growth
My school uses Curriculum Associates' i-Ready adaptive diagnostic to track students' mathematical learning over the course of the year. The full diagnostic assessment, which includes 40 to 50 questions and takes students roughly 90 minutes to complete, is administered three times each year. Students also take 20-question growth-monitoring checks during the interims between the beginning-of-year (BOY), middle-of-year (MOY), and end-of-year (EOY) assessments to ensure that their learning is proceeding apace.
An i-Ready summary of adaptive test structure. Many students are initially confused by the i-Ready assessment because they see some questions involving topics that they have not yet studied. For example, a very advanced sixth grader who gets grade-level questions right may progress up to algebra questions using notation that he or she has never encountered before. Teachers administering the i-Ready diagnostic must prepare students for this sort of challenge and emphasize that encountering questions that seem too difficult is evidence of success rather than failure.
Items on the diagnostic are presented to students as multiple-choice questions with four possible answers. As a result, students' abilities to explain their thinking using appropriate mathematical language are not assessed by the test. This increases the importance of developing alternative ways to evaluate students' mathematical writing; for more information about my approach to that, please visit the page about qualitative evidence of student growth.
Sample i-Ready questions. Each student sees a different set of questions depending on his or her performance on previous questions. Questions come from four major mathematical domains: Numbers and Operations, Algebra and Algebraic Thinking, Measurement and Data, and Geometry.
The validity and reliability of the i-Ready diagnostic have been studied fairly extensively, and the data are encouraging. i-Ready scores tend to correlate closely with success on other standardized assessments like the PARCC, which my students take each spring. This correlation is partially due to i-Ready lessons, which are interactive, computer-based learning modules designed to fill gaps in student learning. Students are automatically assigned these lessons based on their diagnostic scores, and my scholars spend at least an hour each week working on them.
Information from Curriculum Associates about the power of i-Ready testing data. The company demonstrates that the diagnostic is closely aligned with Common Core standards and can be used to predict performance on high-stakes end-of-year tests.
Despite all of the evidence that i-Ready is a strong test, it is worth noting that the margin of error (+/-6 points for the full diagnostic and +/-12 points for the shorter growth-monitoring check) is quite large for a test on which 11 points represents one year of growth for a seventh or eighth grader and 13 points represents one year of growth for a sixth grader. In other words, student levels reported by the diagnostic can be off by roughly half a year in either direction, and student levels reported by the growth-monitoring check can be off by twice as much. This limits educators' abilities to draw meaningful conclusions about individual student growth, particularly if the student in question has only taken the test two or three times.
For example, if a seventh-grade student scores 500 on the BOY diagnostic, his or her score range extends from 494 to 506. If he or she then scores 511 on the MOY diagnostic, one cannot draw any conclusions with certainty. If student's "true" BOY score was 494 and his or her "true" MOY score was 517, he or she grew 23 points, which represents more than two years of learning. However, if his or her "true" BOY score was 506 and his or her "true" MOY score was 505, he or she actually declined by a point.
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Fortunately, this uncertainty can be mitigated dramatically by averaging together groups of student scores. It seems reasonable to assume that errors in score measurement are random (i.e., that any given score is just as likely to be too high as it is to be too low) and independent (i.e., that the error in one student's score is not correlated with the error in any other student's score). If this is the case, the error for a group of scores averaged together can be approximated by the equation below, in which E represents the margin of error for a single student's score and n represents the number of scores that have been averaged together.
Therefore, if I average together the scores of all the students in a single class, the error becomes much smaller. This makes it significantly easier to draw conclusions about student growth. For example, my current seventh-grade class has 18 students, so the margin of error for an averaged class i-Ready diagnostic score would be +/-6 divided by the square root of 18. This results in a margin of error of just +/-1.4 points, which is substantially lower than the margin for error for a single student's score. Indeed, if my class average increases from 500 on the BOY diagnostic to 511 on the MOY diagnostic, I can say with a fair degree of certainty that my class as a whole has grown by at least 8.2 points (because the BOY average could be as high as 501.4 and the MOY average could be as low as 509.6). As my other classes have more students, the margin of error on their average scores is even smaller (+/-1.1 points for the sixth grade and +/-1.2 points for the eighth grade). And, when my 70 students' scores are all averaged together, the margin of error on the figure shrinks to just +/-0.7 points. With these margins of error in mind, consider the table below.
Data table of i-Ready scores from 2016-2017 and 2017-2018 academic years. Rows labeled with numbers show the average scores for that grade level, while rows labeled with "ALL" show average scores for all of my students across the three grades. Remember that 13 points represents a year of growth for sixth grade and that 11 points represents a year of growth for seventh and eighth grade.
There are many things to celebrate among these data. Most noticeably, all of my classes have grown more over the course of the first half of this year than their counterparts grew over the entirety of last year. Already, my current sixth graders have made more than year and a half of growth, my current seventh graders have made nearly two years of growth, and my current eighth graders have made about two and half years of growth. This is very exciting for me as a teacher, and I believe that the dramatic improvement from last year's scores is attributable, at least in part, to improvements I have made to my teaching practice.
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Furthermore, the absolute levels of mastery that my students are showing are impressive. My current seventh and eighth graders are roughly a year and a half beyond the grade-level at which my previous seventh and eighth graders ended the 2016-2017 school year. My current sixth graders' level of mastery is statistically indistinguishable from the point at which my previous sixth graders ended the year, but even being at the same level after only half a year of instruction is quite an accomplishment.
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These data are particularly encouraging because I loop with my students. In other words, last year's sixth graders are this year's seventh graders, and last year's seventh graders are this year's eighth graders. Unfortunately, the point scales for each grade-level are slightly different (making year-to-year comparisons tricky), but the overall pattern is quite clear. My sixth graders arrived last year averaging 470.5 on their diagnostic, which was nearly 25 points below the minimum score (495) at a sixth-grade level. By the end of they year, they had achieved more than a year of growth and averaged 486.4. However, they experienced significant "summer slide," dropping down to an average of 481.2 on their seventh-grade diagnostic. This may sound like relatively little movement, but the minimum grade-level score for seventh graders is a 505. Thus, they were nearly 25 points behind where they needed to be yet again. However, they have grown dramatically over the first half of this year, which suggests that their summer slide was not permanent. Indeed, they are now averaging 502.6, which is extremely close to the grade-level benchmark of 505.
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Last year's seventh graders arrived nearly 35 points behind the 505 grade-level benchmark, made more than a year of growth, and ended within 20 points of a grade-level class average. This year, their 479.9 diagnostic average placed them just over 35 points below the eighth grade benchmark of 515, suggesting that they slid a great deal during the summer too. However, like the current seventh graders, they have make impressive gains over the first half of this year. Their current average score is 507.5, which is just 7.5 points below the grade-level target.
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I did not teach the current sixth graders last year, so their data provide an interesting point of comparison. The two classes I taught last year came back from the summer averaging 11.0 points (a full grade-level) more than their counterparts from the previous year. This suggests that I had a positive impact on their learning (compared to prior teachers) even with the summer slide that they experienced. In contrast, my current sixth graders arrived in my classroom below the level of my 2016-2017 sixth graders and nearly 30 points below their minimum grade-level target. However, this class has also made dramatic growth over the first half of the year, and they are now only 7.9 points away from their 495-point goal.
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Of course, it is difficult to predict the future, and I would be surprised if my if students grew quite as much in the remaining months of the school year as they did at the beginning. As I alluded above, I believe that some of the gains they made during the first half of this year may actually have been the result of teaching that I did last year. Their summer slides were significant, but it appears that they were not "sticky." Students have quickly made up the slippage and grown past where they ended the previous year. However, I also believe that my teaching has improved and continues to do so, so I absolutely expect my students to make additional gains during the second part of the year. Even if I conservatively estimate only half as much growth during the second part of the year, the numbers are impressive.
Data table of EOY Projections for this year. Even a conservative projection of student growth during the second semester suggests that all of my grades will exceed their grade-level targets by the end of this year. On average, this projection shows my students growing three grade levels over the course of the 2017-2018 academic year.
I have intentionally focused on average class scores because they make the data easier to interpret and are more instructive given the i-Ready assessment's large margins of error. However, it is important to demonstrate that scholars across the achievement spectrum are making meaningful gains in my classes. Gains in average scores, after all, are not always the result of uniform progress. If only scholars who started the year with high scores are making large gains, that leaves many others behind. And if score gains come only from low-achieving scholars, I am failing to challenge my high achievers sufficiently. That said, scholars with low BOY diagnostic scores are likely to show the most growth. My school uses the diagnostic scores to determine which students need extra math interventions, so scholars with low initial scores tend to get a great deal of support over the course of the year. This includes extra time on i-Ready lessons, additional small-group instruction, and targeted math practice during our daily enrichment block.
Comparison of growth for scholars with high and low BOY scores. The left chart shows students who scored in the top quartile (i.e., 25%) on their BOY diagnostic assessment, while the right chart shows those who scored in the bottom quartile (i.e., 25%) on their BOY diagnostic assessment. Clearly, most of my scholars have made substantial gains this year. As predicted, though, scholars who performed less well on the BOY assessment and have received additional supports in math have made even more dramatic growth. Still, I am pleased to see that most of my high-achieving scholars are also growing rapidly and are likely to end the year well above grade-level in math.
