In order to evaluate assistive feasibility for assistive mechanism, manipulability magnitude comparison with one direction by one direction is almost inefficient.

Since the manipulability ellipsoid is used to represent the manipulability magnitude on the whole directions, we can compare the ME-al and ME-saam in terms of geometry to obtain the manipulability magnitude comparison results on all the directions.

Table 1 presents additional data on predictive validity of magnitude comparison measures.

Seethaler and Fuchs (2010) administered a single proficiency measure, a magnitude comparison (Chard et al., 2005), and a multiple proficiency measure (the Number Sense, created by the authors) in September and May of kindergarten.

Recent syntheses of the literature on mathematics disabilities (e.g., Desoete, Ceulemans, Roeyers, & Huylebroeck, 2009; Geary, 2004) observe that, in addition to problems with magnitude comparison, counting strategies, and computational strategies, these students often display deficits in working memory (Geary, Hoard, Byrd-Craven, Nugent, & Numtee, 2007; Swanson & Beebe-Frankenberger, 2004) and problems with visual-spatial memory and elaboration (e.g., Geary, 2004).

Both of these are far more comprehensive than measures of one component of number sense, such as magnitude comparison. However, the realities of universal screening require use of the most efficient measures.

TABLE 1 Predictive Validity of Screening Measures for the Primary Grades Study Screening Measure (a) Grade Nh Magnitude Comparison Baglici, Codding, & Test of Early K 61 Tryon (2010) Numeracy (TEN).

Berch (2005) captures the complexities of articulating a working definition of number sense, remarking, "Possessing number sense ostensibly permits one to achieve everything from understanding the meaning of numbers to developing strategies for solving complex math problems; from making simple magnitude comparisons to inventing procedures for conducting numerical operations" (p.

These findings support previous evidence that suggests that DD children are slower to compare numerosities in the subitizing range (Koontz & Berch, 1996) and that difficulties in this processing range can be shared with a normal development of counting and

magnitude comparisons (Bruandet, Molko, Cohen, & Dehaene, 2004).

No differences were found on the Magnitude Comparisons subtest (p = .

Findings revealed that students in the treatment condition outperformed comparison students by .5 of a standard deviation and demonstrated statistically significantly higher scores than comparison students on the TEMIPM Total Score and three of the four subtests (there were no differences between groups on the Magnitude Comparisons subtest).

= standard deviation; TEMI-PM = Texas Early Mathematics Inventories-Progress Monitoring; MC = Magnitude Comparisons subtest; PV = Place Value subtest; ASC = Addition/Subtraction Combinations subtest; NS = Number Sequences subtest; TS = Total Score; TEMI-O = Texas Early Mathematics Inventories-Outcome; MPS = Mathematics Problem Solving subtest; MComp = Mathematics Computation subtest.