Participants were able to demonstrate derived relations by identifying standard formulas and their graphical representations and vice versa, even though the experimental preparations specifically precluded any direct training of these relations. The protocols have employed matching-to-sample (MTS) strategies to teach formula-to-graph relations for mathematical transformations about the coordinate axes ( Ninness, Rumph, McCuller, Vasquez, et al., 2005) and to teach formula-to-factored-formula and factored-formula-to-graph relations for vertical and horizontal shifts on the coordinate axes ( Ninness, Rumph, McCuller, Harrison, et al., 2005).

Ninness and colleagues have developed several computer-interactive protocols directed at establishing advanced math skills via derived stimulus relations. A frame of opposition refers to stimuli that can be contrasted along some dimension in which objects or events have some order (e.g., opposite of, inverse of, reciprocal of). As in equivalence terminology, a frame of coordination refers to stimuli being the same as or equivalent to other stimuli a frame of comparison involves relating stimuli along some dimension of quantity or quality (e.g., larger than, higher than). Relational frame theory (RFT) extends the equivalence paradigm by suggesting that verbally competent participants can learn to respond in accordance with sameness and other more complex relations among stimuli ( Hayes et al., 2001). The emergence of untrained relations, termed stimulus equivalence ( Sidman, 1994) or relational framing when the stimuli are not directly equivalent (e.g., more than, less than Hayes, Barnes-Holmes, & Roche, 2001), represents a possible advantage for teachers with limited time for math instruction. Also, Lynch and Cuvo (1995) illustrated that teaching fifth and sixth graders to match fraction ratios to their corresponding graphical representations and graphs to the corresponding decimal values resulted in the emergence of nontrained skills to match decimal to fraction and fraction to decimal without direct training. Notwithstanding, innovative behavioral research conducted by Mayfield and Chase (2002) confirmed the beneficial effects of cumulative practice on the acquisition of math skills among low-performing college students. Despite recent legislative attempts such as No Child Left Behind (2001), student performance has failed to show significant improvement. students encounter in their required college course work ( Conway & Sloane, 2005). Japanese high-school students acquire more advanced math skills (e.g., quadratic functions, permutations and combinations, trigonometric functions, limits, derivatives, and the applications of integrals) than almost all U.S. In the highest achieving Asian and European countries, eighth-grade mathematics curricula include the basics of three-dimensional geometry, proportionality problems, and transformation of geometric functions (see Schmidt, Houang, & Cogan, 2002, for a detailed discussion). Insufficient exposure to mathematics instruction may account for some of these findings. students are asked to apply what they have learned in mathematics, they demonstrate less ability than most of their peers in other highly industrialized countriesРІР‚Сњ (PISA, 2006, as cited in U.S. These mathematical difficulties were also reflected in the most recent PISA outcomes on science literacy, which indicate that РІР‚Сљwhen older U.S. students proving to be incapable of responding to items that require fundamental algebra and the most rudimentary levels of basic computations ( PISA, 2003). The most troubling areas in the 2003 evaluation were math literacy and problem solving, with U.S. 10th-grade students in all quantitative skill areas. high-school students can only be described as discouraging, with nearly every developed nation surpassing U.S.

The Programme for International Student Assessment (PISA), a worldwide appraisal of high-school student performance, makes it obvious that the performance levels of U.S. high-school students have consistently ranked lower on math literacy than many industrialized and nonindustrialized countries (24 of 29 countries) in the Organization for Economic Cooperation and Development (OECD). International indicators of mathematical performance suggest that the mathematical skills of high-school and entry-level college students in the U.S.

Discrete mathematics ensley 2006