1. Spatial and Numerical Predictors of Measurement Skills in Boys and Girls from Lower- and Higher-Income Communities Beth Casey, Marina Vasilyeva, & Eric Dearing Boston College

2. Why study measurement? A foundational concept for learning science One of the most widely used applications of math A key conceptual construct in math Provides a link between spatial and numerical reasoning Bridges two main areas of math – geometry and number

3. Measurement is one of the most challenging domains of elementary school math Large-scale national and international studies: Largest achievement gaps in measurement skills –SES (low-income < high- income) –Gender (girls < boys) Need to move beyond simple documentation of group differences Critical to understand factors contributing to these differences NAEP (National Assessment of Educational Progress) TIMMS (Third International Mathematics and Science Study)

4. Two key subtypes of measurement To systematically examine individual differences, we developed a measurement assessment tool Covering a wide range of measurement concepts introduced in elementary school Including length, perimeter, scaling, area, volume Two subtypes of measurement skills were identified through factor analysis Formula-based items –Can be solved analytically, using known measurement formulas –Spatial reasoning is not required but can be helpful Spatial/conceptual items –Cannot be solved simply by relying on formulas –Require an understanding of the spatial relations underlying measurement procedures »Subdividing space into equal parts and visualizing unit structures »Forming and manipulating mental images

5. Research questions Do spatial and numerical skills predict measurement performance? For formula-based items For spatial/conceptual items Does the pattern of relations between the predictors and the two subtypes of measurement vary as a function of income level? What is the nature of the relation between gender and measurement performance for each income group? Is this relation mediated by numerical and/or spatial skills?

6. Method Participants 190 fourth-graders (93 girls and 97 boys) –91 from higher-income community –99 from lower-income community Participants across communities received regular math instruction with the same curriculum Materials and procedure Three testing instruments were group-administered »Measurement test (day 1) »Numeric test (day 2) »Spatial visualization test (day 2)

7. Instruments Measurement test 34 multiple-choice items –14 formula-based –20 spatial/conceptual

8. 8 square inches 8 square inches Find the AREA of this shape. 16 square inches 16 square inches 15 square inches 15 square inches 30 square inches 30 square inches 3 inches 5 inches Example 1 of formula-based measurement item

9. 16 inches 8 inches 4 inches 3 inches Cristina drew a rectangle. The length of the rectangle is 8 inches. The perimeter of the rectangle is 24 inches. What is the WIDTH of the rectangle in inches? Example 2 of formula-based measurement item

10. 20 feet 12 feet 7 feet 5 feet John made four guesses about the HEIGHT of the door (in feet) to his bedroom. Which one do you think is the best guess? Example 1 of spatial/conceptual measurement item

11. 30 tiles 15 tiles 60 tiles 35 tiles Yang measured the area of the room by covering the floor with tiles of size A below. Rachel measured the same room by covering the floor with tiles of size B below. Yang used 30 tiles. How many tiles did Rachel use? size A size B Example 2 of spatial/conceptual measurement item

12. Instruments (con’t) Numeric test (14 multiple-choice items) Number facts and word problems Covering all four arithmetic operations Based on TIMMS and NAEP items for 4 th graders »E.g., “How much is 42-29?” Spatial visualization test (8 puzzles) Adapted from the Jigsaw-Puzzle Imagery task (Richardson & Vecci, 2002) Drawing of a common object cut into equal-sized rectangular puzzle pieces Participants not allowed to physically move puzzle pieces Task: Identify correct location of puzzle pieces on empty grid to re-create the object.

14. Results Accuracy of performance on the two subtest of measurement (percent correct) Spatial/conceptual Items Formula-based Items Higher- income 77% (SD = 15%) 67% (SD = 24%) Lower- income 40% (SD = 17%) 36% (SD = 20%)

15. Path Analysis Plan A series of path analysis models were run Outcome variables –formula-based measurement scores –spatial/conceptual measurement scores Cognitive predictor variables –Numeric test scores –Spatial visualization scores We compared models, in which patterns of associations were assumed to be equal for the two income groups patterns of associations were allowed to differ between income groups The best model fit was obtained when relations between cognitive predictors and outcome variables were allowed to vary as a function of income level For each income group, we then examined direct and indirect associations between gender and outcome variables

16. Path Analysis Results for Higher-income Students Spatial/conceptual measurement was predicted by both numeric and spatial visualization skills (41% of variance explained) Formula-based measurement was also predicted by both numeric and spatial visualization skills (44% of variance explained) There were no significant gender differences on either subtype of measurement performance No direct effects No indirect effects via numeric or spatial visualization skills

17. Χ 2 = 9.80 df =8, p=.28; NNFI =.98; RMSEA =.05 Spatial Skills Numerical Skills Spat/ Concpt Measurement.26.46.54 b. Gender -.24.37.23 Spatial Skills Numerical Skills Spat/ Concpt Measurement Form.-based Measurement.45.59.42 a..32 Gender.48 Form.-based Model for higher-income group

18. Path Analysis Results for Lower-income Students Similar to higher-income group, spatial/conceptual measurement was predicted by both numeric and spatial visualization skills (42% of variance explained) In contrast to higher-income group, formula- based measurement was predicted only by numeric skills (28% of variance explained) There were both direct and indirect effects of gender

19. Gender Effects in Lower-Income Students Gender differences Boys performed significantly better than girls on spatial/conceptual measurement items Girls performed significantly better than boys on formula- based measurement items Girls performed significantly better than boys on numeric test Gender directly associated with spatial/conceptual subtest Boys’ had a direct advantage on spatial/conceptual measurement items Gender indirectly associated with formula-based subtest Numeric skills mediated girls’ advantage on formula-based measurement items

20. Χ 2 = 9.80 df =8, p=.28; NNFI =.98; RMSEA =.05 Spatial Skills Numerical Skills Spat/ Concpt Measurement.26.46.54 b. Gender -.24.37.23 Spatial Skills Numerical Skills Spat/ Concpt Measurement Form.-based Measurement.45.59.42 a..32 Gender.48 Form.-based Model for higher-income group Model for lower-income group

21. Discussion Striking differences between lower- and higher-income students: –A very different level of performance on measurement –A different pattern of relations among predictors for higher- and lower-income students –Differences in gender effects

22. Higher-Income Students Higher-income students seem to have synthesized spatial and numeric understanding of measurement Able to draw on both types of cognitive skills when solving measurement problems across the board –Spatial understanding of measurement comes in useful even when deciding what formula to apply Possible alternative interpretation of findings General intelligence factors, e.g., brighter students did better on all measures However, when controlled for verbal IQ, numerical and spatial skills still significantly predicted for measurement

23. Higher-Income Students Higher-income students seem to have synthesized spatial and numeric understanding of measurement Able to draw on both types of cognitive skills when solving measurement problems across the board –Spatial understanding of measurement comes in useful even when deciding what formula to apply Possible alternative interpretation of findings General intelligence factors, e.g., brighter students did better on all measures However, when controlled for verbal IQ, numerical and spatial skills still significantly predicted for measurement

24. Higher-Income Students Lack of gender differences on both subtests of measurement contrasts with general findings of a male advantage reported in large-scale studies Both boys and girls performed quite well Girls on average scored nearly 80% correct on the spatial/conceptual items These girls clearly were not having major difficulties with the items requiring spatial thinking

25. Lower-income students In contrast to higher-income students, lower- income students did not appear to use their “spatial sense” when solving formula-based problems Only the numeric test predicted their performance on these items Apparently, they approached the two subtypes of measurement problems using different strategies May not understand the connections between measurement formulas and underlying spatial relations

26. Gender differences in lower-income students Girls’ higher performance on formula-based items was mediated by their numeric skills advantage Female advantage on the numeric test is consistent with the gender differences on computational skills reported in the literature Female numeric advantage found in the lower-income group in the present study is also consistent with the finding in the literature that the female advantage occurs more frequently among students at below-average skill levels (Martin & Hoover, 1987) Girls' numerical advantage benefited their performance on the spatial/conceptual items But, it did not compensate sufficiently to overcome the direct advantage that the boys had on the spatial/conceptual items

27. Gender differences in lower-income students Lower-income boys had a better conceptual understanding of spatial/conceptual items -- the measurement items that could not be solved through use of formulas Boys’ performance on spatial/conceptual measurement was not mediated by their visualization skills. Therefore, visualization skills cannot explain the male advantage on this type of measurement Note that spatial visualization skills predicted for spatial/conceptual measurement items for both boys and girls Alternative explanation for the male advantage: Boys’ advantage may be due to a greater familiarity with measurement tools and a greater understanding of measurement units –For example, in estimating the height of a door, the boys were more accurate and less likely than the girls to choose the improbable answer of 20ft

28. Educational implications In the present study, both lower- and higher-income school systems used the same math curriculum (Investigations in Number, Data, and Space) NCTM standards-based Strong spatial emphasis »For example, the concept of area is introduced by having students construct spatial arrays rather than focusing on formula-based solutions

29. Investigations in Data, Number and Space: A lesson on area

30. Educational implications In the higher-income school system, girls as well as boys appear to have been successfully taught how to use spatial reasoning skills to solve measurement problems Hence: –lack of gender differences –high level of performance on measurement –integration of spatial and numeric reasoning This hypothesis needs to be examined systematically in future research comparing spatially-focused and non-spatial curricula

31. Educational implications Why have the lower-income students not benefited as much from the spatially-focused curriculum? Possible explanations: –Teacher effects: teachers in the higher-income community may have been more effective and skillful at using this curriculum –Peer effect: differing ability levels and knowledge base of their classmates –Home effects: large differences in home environments in: »general level of cognitive stimulation »input relating to math and measurement concepts »specific assistance on math and measurement homework

32. Conclusions Present findings on lower-income students and national assessments show that students perform poorly on measurement content Typically, little time is spent on measurement instruction in elementary school math Obvious conclusion: greater attention needs to be paid to this critical aspect of mathematics

33. Conclusions A number of educational models for teaching measurements skills primarily focus on formula solutions, while others focus on more conceptual understanding Even when both approaches are taught within the same curriculum, they are often taught in isolation from one another, with no clear connection made between them A bridge needs to be made between these two approaches to measurement This link would enable students to recognize that they can utilize and combine strategies, drawing on both formula- based knowledge and spatial/conceptual understanding when solving the same measurement problem