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NAEP Instruments → Contextual Items (Survey Questionnaires) → Indices Based on Questionnaire Data → Development of the 2017 Grade 4 Mathematics Indices

NAEP Technical Documentation Development of the 2017 Grade 4 Mathematics Indices

For the 2017 grade 4 mathematics assessment, several indices of policy interest were developed that satisfied both theoretical criteria based on content analysis, and empirical criteria based on multivariate statistical techniques. This resulted in the creation of several new reporting elements. The development of the 2017 grade 4 mathematics indices can be summarized in three main steps:

1. Question development. New sets of contextual items, such as those exploring students’ persistence in learning and their enjoyment of complex problems, were developed and included in the mathematics student questionnaire. Through content analysis as part of the item development process, only sets of items that were theoretically interpretable and meaningful as a conceptual unit were included as potential indices to measure specific constructs of interest.

2. Examination of empirical relationships. Factor analysis was used to explore and verify the empirical properties of the data. Construct validity of the potential indices was evaluated through factorial validity with respect to the survey question responses, and the convergent and discriminant validity of the factor with respect to other factors. If the factor had the expected pattern of relationships and non-relationships, the construct validity of the factor as representing the intended index was supported.

3. Index scoring. The partial credit item response theory (IRT) model was used to scale the indices. Scaling of the index items was first conducted to get the item parameters and was based on the marginal maximum likelihood methodologies. After the parameters were estimated, expected a posteriori (EAP) scores were calculated as the estimate of the index score. Then, the EAP scores were transformed to have a mean of 10 and a standard deviation of 2, and were reported on a scale from 0–20.

Index of Students' Persistence in Learning

The table below presents the items forming the index of students' persistence in learning. This index was designed to measure students' tendency to persevere and work hard in the face of challenges. Grade 4 students were asked to indicate how much each of the four statement items described a person like them (not at all like me, a little bit like me, somewhat like me, quite a bit like me, or very much like me).

Items in the index of fourth-grade students' persistence in learning, grade 4 mathematics combined national and state assessment: 2017
How much does each of the following statements describe a person like you? Select one answer choice on each row.
Response categories
Item			Not at all like me	A little bit like me	Somewhat like me	Quite a bit like me	Very much like me
B034901	a.	I finish whatever I begin.	A	B	C	D	E
B034902	b.	I try very hard even after making mistakes.	A	B	C	D	E
B034903	c.	I keep working hard even when I feel like quitting.	A	B	C	D	E
B034904	d.	I keep trying to improve myself, even when it takes a long time to get there.	A	B	C	D	E
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 2017 Mathematics Assessment.

Index of Students' Enjoyment of Complex Problems

The table below presents the items forming the index of students' enjoyment of complex problems. This index was designed to measure students' enjoyment of problems and activities that challenge them to think. Grade 4 students were asked to indicate how much each of the four statement items described a person like them (not at all like me, a little bit like me, somewhat like me, quite a bit like me, or very much like me).

Items in the index of fourth-grade students' enjoyment of complex problems, grade 4 mathematics combined national and state assessment: 2017
How much does each of the following statements describe a person like you? Select one answer choice on each row.
Response categories
Item			Not at all like me	A little bit like me	Somewhat like me	Quite a bit like me	Very much like me
B035101	a.	I like complex problems more than easy problems.	A	B	C	D	E
B035102	b.	I like activities that challenge my thinking abilities.	A	B	C	D	E
B035103	c.	I enjoy situations where I will have to think about something.	A	B	C	D	E
B035104	d.	I enjoy thinking about new solutions to problems.	A	B	C	D	E
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 2017 Mathematics Assessment.

Index of Students’ Confidence in Mathematics Knowledge and Skills

The table below presents the items forming the index of students' confidence in their mathematics knowledge and skills at grade 4. This index was designed to measure students' belief in their abilities to do various mathematics-related tasks. Grade 4 students were asked to indicate their confidence in doing the task described in each of four items (I definitely can’t, I probably can’t, maybe, I probably can, or I definitely can).

Items in the index of fourth-grade students' confidence in mathematics knowledge and skills, grade 4 mathematics combined national and state assessment: 2017
Thinking about math, do you think that you would be able to do each of the following? Do not actually solve the problems. Select one answer choice on each row.
Response categories
Item			I definitely can't	I probably can't	Maybe	I probably can	I definitely can
M831401	a.	Estimate the weight of 5 apples using pounds (lbs.)	A	B	C	D	E
M831402	b.	Divide 42 stickers among 6 students	A	B	C	D	E
M831405	c.	Find the amount of carpet needed to cover a rectangular floor if you know its length and width	A	B	C	D	E
M831406	d.	Know when to take a turkey out of the oven if it goes in at 10:00 A.M. and it takes 3 hours and 45 minutes to cook	A	B	C	D	E
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 2017 Mathematics Assessment.

For each index item, the response categories were scored as numerical values (e.g., for an item with five response categories, category A was scored as 1, B was scored as 2, C was scored as 3, D was scored as 4, and E was scored as 5). For the 2017 grade 4 mathematics indices, item response categories were collapsed; scores for a five-category item thus became 1, 2, 3, and 4 after collapsing. The table below describes the treatment of the index items.

Recoding of polytomous items for scaling, grade 4 mathematics combined national and state assessment, by index: 2017
Item	Index	Reason for decision	Disposition
B034901	Persistence in learning	To improve model-data fit	Collapse categories: 1,2,3,4,5 becomes 1,1,2,3,4
B034902	Persistence in learning	To improve model-data fit	Collapse categories: 1,2,3,4,5 becomes 1,1,2,3,4
B034903	Persistence in learning	To improve model-data fit	Collapse categories: 1,2,3,4,5 becomes 1,1,2,3,4
B034904	Persistence in learning	To improve model-data fit	Collapse categories: 1,2,3,4,5 becomes 1,1,2,3,4
B035101	Enjoyment of complex problems	To improve model-data fit	Collapse categories: 1,2,3,4,5 becomes 1,1,2,3,4
B035102	Enjoyment of complex problems	To improve model-data fit	Collapse categories: 1,2,3,4,5 becomes 1,1,2,3,4
B035103	Enjoyment of complex problems	To improve model-data fit	Collapse categories: 1,2,3,4,5 becomes 1,1,2,3,4
B035104	Enjoyment of complex problems	To improve model-data fit	Collapse categories: 1,2,3,4,5 becomes 1,1,2,3,4
M831401	Confidence in mathematics knowledge and skills	To improve model-data fit	Collapse categories: 1,2,3,4,5 becomes 1,1,2,3,4
M831402	Confidence in mathematics knowledge and skills	To improve model-data fit	Collapse categories: 1,2,3,4,5 becomes 1,1,2,3,4
M831405	Confidence in mathematics knowledge and skills	To improve model-data fit	Collapse categories: 1,2,3,4,5 becomes 1,1,2,3,4
M831406	Confidence in mathematics knowledge and skills	To improve model-data fit	Collapse categories: 1,2,3,4,5 becomes 1,1,2,3,4
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 2017 Mathematics Assessment.

The partial credit IRT model was used to scale the indices. Scaling of the index items was first conducted to get the item parameters and was based on the marginal maximum likelihood methodologies. The following tables show the IRT parameters for the 2017 grade 4 mathematics indices.

IRT parameters for items in the index of fourth-grade students' persistence in learning, grade 4 mathematics combined national and state assessment: 2017
Item	b_j	d_j1	d_j2	d_j3	d_j4	d_j5
B034901	-0.13	0.31	0.15	-0.47	†	†
B034902	-0.67	0.19	0.11	-0.30	†	†
B034903	-0.58	0.23	0.07	-0.30	†	†
B034904	-0.82	0.16	0.03	-0.19	†	†
† Not applicable. NOTE: The number of d_ji parameters is one less than the number of categories for the item. For items scaled with the partial credit model, parameters a and c are not estimated. The a parameter value is exactly one and the c parameter is exactly zero. For item j, b_j represents a location parameter related to item difficulty, and d_ji represents the category threshold parameter for category i of item j; d_ji may not sum to zero because of rounding. SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 2017 Mathematics Assessment.

IRT parameters for items in the index of fourth-grade students' enjoyment of complex problems, grade 4 mathematics combined national and state assessment: 2017
Item	b_j	d_j1	d_j2	d_j3	d_j4	d_j5
B035101	0.23	0.28	0.05	-0.33	†	†
B035102	-0.29	0.24	-0.01	-0.23	†	†
B035103	-0.06	0.30	0.02	-0.33	†	†
B035104	-0.24	0.18	0.03	-0.22	†	†
† Not applicable. NOTE: The number of d_ji parameters is one less than the number of categories for the item. For items scaled with the partial credit model, parameters a and c are not estimated. The a parameter value is exactly one and the c parameter is exactly zero. For item j, b_j represents a location parameter related to item difficulty, and d_ji represents the category threshold parameter for category i of item j; d_ji may not sum to zero because of rounding. SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 2017 Mathematics Assessment.

IRT parameters for items in the index of fourth-grade students' confidence in mathematics knowledge and skills, grade 4 mathematics combined national and state assessment: 2017
Item	b_j	d_j1	d_j2	d_j3	d_j4	d_j5
M831401	0.15	0.89	-0.14	-0.75	†	†
M831402	-0.61	0.27	-0.17	-0.11	†	†
M831405	-0.11	0.58	-0.13	-0.45	†	†
M831406	-0.19	0.23	-0.11	-0.12	†	†
† Not applicable. NOTE: The number of d_ji parameters is one less than the number of categories for the item. For items scaled with the partial credit model, parameters a and c are not estimated. The a parameter value is exactly one and the c parameter is exactly zero. For item j, b_j represents a location parameter related to item difficulty, and d_ji represents the category threshold parameter for category i of item j; d_ji may not sum to zero because of rounding. SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 2017 Mathematics Assessment.

After the parameters were estimated, EAP scores were calculated as the estimate of the index score. Then, the EAP scores were transformed to have a mean of 10 and a standard deviation of 2, and were reported on a scale from 0–20 (transformed scale score = 2 * (index value) + 10). The tables below show the response averages, index values, and transformed scale scores for each of the three grade 4 mathematics indices. Note that response averages represent raw response averages after collapsing. Each response average corresponds to one index value and transformed score.

As a reporting aid, index scores were divided into a range of categories or classifications (e.g., low, moderate, high). The cut points selected to divide the index scores into meaningful categories were based on the distributions of the EAP scores; there are no theoretical derivations behind the selection of the cut points. For example, for the index of students’ persistence in learning, index scores were divided into low, moderate, and high classifications, such that grade 4 students with high index scores (i.e., a transformed score of 9.6 or higher on a scale of 0 to 20) generally reported that statements about persevering and working hard in the face of challenges described a person like them quite a bit or very much. Grade 4 students with low scores on the index (i.e., lower than 8.3) generally reported that these statements described a person like them not at all or a little bit. Higher index values and transformed scores correspond to higher response averages, and lower index values and transformed scores correspond to lower response averages.

As an example, for the index of persistence in learning, grade 4 students were classified as follows:

Students with index scores associated with a response average less than 2 (index value -0.83874) were classified as low on the index. That is, students who on average responded closest to not at all like me on a question with the five response options not at all like me, a little bit like me, somewhat like me, quite a bit like me, and very much like me were classified as having a low level of persistence in learning.
Students with index scores associated with a response average greater than or equal to 2 to less than 3 (index value -0.18397) were classified as having a moderate level of persistence in learning.
Finally, students with index scores associated with a response average greater than or equal to 3 were classified as high on the index. That is, students who on average responded closest to very much like me were classified as having a high level of persistence in learning.

Response averages, index values, and transformed scores for the index of fourth-grade students' persistence in learning, grade 4 mathematics combined national and state assessment, by classification: 2017
Classification	Response average	Index value	Transformed score	Percentage of students
Low	1.00	-1.85057	6.3	2
	1.25	-1.47396	7.1	2
	1.50	-1.21422	7.6	2
	1.75	-1.01194	8.0	4
Moderate	2.00	-0.83874	8.3	5
	2.25	-0.67908	8.6	5
	2.50	-0.52265	9.0	8
	2.75	-0.36080	9.3	7
High	3.00	-0.18397	9.6	10
	3.25	0.02109	10.0	13
	3.50	0.27666	10.6	12
	3.75	0.62628	11.3	14
	4.00	1.16305	12.3	17
NOTE: Detail may not sum to totals because of rounding. SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 2017 Mathematics Assessment.

Response averages, index values, and transformed scores for the index of fourth-grade students' enjoyment of complex problems, grade 4 mathematics combined national and state assessment, by classification: 2017
Classification	Response average	Index value	Transformed score	Percentage of students
Low	1.00	-1.52218	7.0	7
	1.25	-1.08784	7.8	5
	1.50	-0.79958	8.4	5
	1.75	-0.58268	8.8	8
Moderate	2.00	-0.40252	9.2	8
	2.25	-0.24095	9.5	7
	2.50	-0.08668	9.8	10
	2.75	0.06909	10.1	7
High	3.00	0.23523	10.5	8
	3.25	0.42343	10.9	9
	3.50	0.65267	11.3	8
	3.75	0.95949	11.9	7
	4.00	1.42264	12.9	11
NOTE: Detail may not sum to totals because of rounding. SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 2017 Mathematics Assessment.

Response averages, index values, and transformed scores for the index of fourth-grade students' confidence in mathematics knowledge and skills, grade 4 mathematics combined national and state assessment, by classification: 2017
Classification	Response average	Index value	Transformed score	Percentage of students
Low	1.00	-1.69633	6.6	3
	1.25	-1.26696	7.5	3
	1.50	-0.96797	8.1	3
	1.75	-0.73546	8.5	7
Moderate	2.00	-0.53801	8.9	10
	2.25	-0.35828	9.3	8
	2.50	-0.18492	9.6	11
	2.75	-0.00876	10.0	9
High	3.00	0.17959	10.4	10
	3.25	0.39222	10.8	10
	3.50	0.64792	11.3	9
	3.75	0.98127	12.0	8
	4.00	1.46412	12.9	8
NOTE: Detail may not sum to totals because of rounding. SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 2017 Mathematics Assessment.

Last updated 05 August 2022 (ML)

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