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Reasoning vs. Extracting Clues from Visual Imageries Although in the previous section we offered a description of the type of tasks that demanded abstract and generalized thinking, a closely linked major difference between the new national items at all grade levels and those on previous state exams included the level of abstraction and formal knowledge expected of children when solving problems. Although the state items often included images, representations, pictures, and models from which the children could extract information to solve problems, the new national items either did not provide such visual clues or when they did, the images could not be used directly to find answers. On state items, most typically, when children were asked to find the area, perimeter, or volume of geometric shapes, those shapes were presented on a grid where children could count to obtain answers. Additionally, whole numbers were often used rather than fractions. Such was not the case for similar items on the new national assessments. Notice that in the example showing in Figure 5, children need to alter or manipulate the accompanied image to answer the question. Notice also that the visual media presented to children does not include features that could be used to solve the problem without understanding the concept of perimeter. Indeed, knowing the formula alone would not be sufficient to solve the tasks. Vol. 3, No. 4 Manouchehri et al.: What Do Mathematics Achievement... 326 Figure 5. National Grade 4 Item #29 Source: Retrieved from bit.ly/22rtSrM. We do stress that the analysis offered in this paper was based solely on the available practice items from one of the two assessment consortia creating the new national assessments, and on items from released state sample tests. The released tests did not include all open response items or performance-based tasks that are used. Any inferences about the state tests or national assessments must be cautionary due to the fact that neither full examinations nor an accurate distribution of items according to content strands was made available. The primary value of analysis is to help identify new types of knowledge children would need to master on the new national assessments. It is often said that in educational settings what is tested is what is taught. If that is indeed the case, and if the released national practice items are a representative of what knowledge is to be valued, then there is some indication that teachers must help children develop skills consistent with mathematical practices so to successfully meet the demands of the new assessments. As discussed in previous sections, while no significant differences existed among the achievement tests used by different states and the new national released achievement items according to the cognitive load they demanded of children, Athens Journal of Education November 2016 327 the degree of decision-making and abstract reasoning expected of children was remarkably higher than the standardized state achievement exams. Certainly both in quality and in quantity, the questions that measured abstract reasoning, structural knowledge, and tending to precision ranked higher on the new national items as compared to the prior state items. As we mentioned earlier, questions addressing some of the content strands were not included among the items designed for certain grade levels on the new national assessment items, or they appeared less frequently. For instance, Geometry items were limited to only 4 in the fourth grade. However, in reviewing fifth grade Geometry items it is clear that meeting the mathematical expectations of knowledge of Geometry at that grade may not be met without adequate preparation in the previous years. The same applies to the Data Analysis/Probability items appearing in Grade 6. A large number of the new national assessment items require students to do a substantial degree of abstracting. They also demand that students be flexible problem solvers. The skills needed for student success on these items require the skills to be built over time and through the use of activities and lessons that focus on the development of mathematical thinking among children. Attempts at educating mathematics teachers must capitalize on the demands of these tasks and engage them in explicit discussions focused on understanding the mathematical connections that need to be unveiled in instruction.

What Do Mathematics Achievement Examinations Assess? A Critical Item Analysis