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Heuristic and Algorithmic Processingin English, Mathematics, and ScienceEducationMatthew J. Sharps a , Adam B. Hess b , Jana L. Price-Sharps c &Jane Teh aa California State University, Fresnob California State University, Fresnoc Alliant International UniversityPublished online: 07 Aug 2010.

To cite this article: Matthew J. Sharps , Adam B. Hess , Jana L. Price-Sharps & Jane Teh(2008) Heuristic and Algorithmic Processing in English, Mathematics, and Science Education,The Journal of Psychology: Interdisciplinary and Applied, 142:1, 71-88, DOI: 10.3200/JRLP.142.1.71-88

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7171

Heuristic and Algorithmic Processing in English, Mathematics,

and Science Education

MATTHEW J. SHARPSCalifornia State University, Fresno

ADAM B. HESSCalifornia State University, Fresno

Alliant International University

JANA L. PRICE-SHARPSAlliant International University

JANE TEHCalifornia State University, Fresno

ABSTRACT. Many college students experience difficulties in basic academic skills. Recent research suggests that much of this difficulty may lie in heuristic competency—the ability to use and successfully manage general cognitive strategies. In the present study, the authors evaluated this possibility. They compared participants’ performance on a prac-tice California Basic Educational Skills Test and on a series of questions in the natural sciences with heuristic and algorithmic performance on a series of mathematics and read-ing comprehension exercises. Heuristic competency in mathematics was associated with better scores in science and mathematics. Verbal and algorithmic skills were associated with better reading comprehension. These results indicate the importance of including heuristic training in educational contexts and highlight the importance of a relatively domain-specific approach to questions of cognition in higher education.

Keywords: algorithms, California Basic Educational Skills Test, cognition, education, gestalt/feature-intensive processing, heuristic competency

MANY EDUCATORS AT ALL GRADES AND LEVELS report increasing dif-ficulty in teaching both basic and advanced skills. Test results even at collegiate levels are less than encouraging, exemplified strongly in the case of the California

Address correspondence to Matthew J. Sharps, Department of Psychology, MS ST-11, California State University, Fresno, CA 93740-8019, USA; matthew_sharps@csufresno.edu (e-mail).

The Journal of Psychology, 2008, 142(1), 71–88Copyright © 2008 Heldref Publications

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Basic Educational Skills Test (CBEST), recently used to qualify teachers for the classroom in California (e.g., Ahlquist, Berlak, Castaneda, Lea, & Montano, 2004). Large numbers of students attempting to qualify as teachers have been unable to do so in the areas addressed by the CBEST, which are “basic English literacy, writing, and math” (Ahlquist et al., p. 15). The problem has proven particularly severe for minority teaching candidates. Although Kelemen and Koski (1998) estimated that about 75% of candidates passed the CBEST on the first attempt, Ahlquist et al. estimated that as many as 62% of African Americans, 50% of Latinos, and 47% of Asian Americans, together with approximately 20% of European Americans, have been eliminated as candidates for teaching careers, at least in California, by their inability to demonstrate basic competencies in these areas on the CBEST. This lack of basic competencies poses obvious and important problems that need to be addressed, including the fact that so many candidates for teaching careers, regard-less of ethnicity, are so poorly prepared in basic English and math skills throughout their college careers.

Basic mathematics and language skills are not the only areas exhibiting puzzlingly diminished performance. Scholars also perceive science education, in terms of criteria such as achievement levels and students enrolled in science curricula, to have become increasingly problematic in recent years for a variety of well-documented reasons (e.g., Teitelbaum, 2003). However, researchers have not yet thoroughly explored the important relation between verbal and math-ematical skills and science education at a quantitative level.

What are the reasons that so many evidently qualified college graduates are unable to pass the CBEST? This question is encapsulated in a somewhat larger inquiry: What are the cognitive dynamics underlying advanced students’ prob-lems with (a) basic skills in language and mathematics and (b) the skills needed for effective science education? What, cognitively, is lacking?

Basic algorithmic skills would not seem to be a likely source of these difficul-ties. Algorithms are predetermined procedures or sets of instructions for carrying out given operations in a finite numbers of steps (e.g., Gregory, 1987). The learning of a simple number of steps to perform a purely algorithmic mathematical operation, such as carrying in multiplication, or to perform a purely algorithmic verbal operation, such as extracting explicit meaning from an explicit sentence (e.g., “Mary had a blue car. What color was Mary’s car?”) requires only basic language skills and an under-standing of the concrete steps involved. It seems unlikely that teaching candidates’ CBEST performance or college-level skills in math, language, or science would be enhanced through the provision of additional rote knowledge in these areas.

However, the heuristic realm is another matter. The concept of the heuristic in older literature was initially construed simply as a mental technique that increased the probability of solution, as opposed to the concept of the algorithm, which in principle guarantees a solution. This early definition leads to immediate difficul-ties; by this definition, effectively, a heuristic could be construed as any mental technique that can be applied positively to a problem and is not an algorithm.

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However, scholars have refined and evaluated the idea of heuristic process-ing through several decades of behavior decision research theory into a much richer concept embedded in larger issues of cognitive processing (e.g., Bazer-man, 1998; Camerer, 1995; Cialdini, 1988; Denney, 1989, 1990; Freedman & Fraser, 1966; Garnham & Oakhill, 1994; Gilovich, 1992; Kahneman & Tversky, 1972, 1979; Kintsch, 1979, 1994; Medin & Bazerman, 1999; Payne, 1973, 1982; Reese & Rodeheaver, 1985; Simon, 1957; Tversky & Kahneman, 1972, 1973, 1974). This long-term refinement has been congruent with increasing interest in cognitive studies of the importance of contextual processes in real-world deci-sion making (e.g., Blanchard-Fields, 1986; Gauvain, 1993; Kohler, 1947; Medin & Bazerman; Park, 1992; Sharps & Wertheimer, 2000; Wertheimer, 1982; Willis, 1991, 1995, 1996), as we address in the current study with reference to the types of academic performance measured by the CBEST.

Cognitive processing in real-world contexts has been known for several decades to be more heuristically than algorithmically based (e.g., Newell & Simon, 1972), and, thus, heuristic processing has rightly taken center stage for several decades in the search for better characterizations of cognition in applied settings. Heuristics, broadly speaking and in textbook terms, are general cogni-tive strategies that are typically accurate (e.g., Matlin, 2005); as such, they reduce cognitive strain (Bourne, Dominowski, & Loftus, 1979). Although they may, as is occasionally still asserted (e.g., Gregory, 1987), be based to some degree on trial and error, typically the most important factors in heuristic processing lie in meaning—in the understanding, based on probabilistic experience and the inter-pretation of that experience, of the context, parameters, and boundaries of a given problem space or situation (e.g., Kohler, 1947; Sharps & Martin, 2002; Sharps & Wertheimer, 2000; Wertheimer, 1982). Heuristics may operate implicitly, such as when a given problem space in the mathematical or verbal realm is processed with reference to possible solution sets or strategies that have been used in the past to address problems with similar identified features.

Central to the success of a given heuristic strategy is the grasping of core, meaningful features of the problem space, termed the radix or root by the early Gestaltists (e.g., Kohler, 1947; Sharps & Wertheimer, 2000; see also Wertheimer, 1982). The basic concept is that if the context of a cognitive task is understood, meaningful solutions will be forthcoming; if not, solutions are likely to be mind-less, wrong, or too narrow to be useful (e.g., Grisso, 1986; Kohler; Lawton, 1982; Sharps & Martin, 2002; Sharps & Wertheimer; Wertheimer).

General, probabilistic, loose-fit problem-solving and information-processing strategies, which are based on prior experience and enable the respondent to extract root concepts pertinent to the most probable boundaries of a problem space, form the core of heuristic and gestalt processing in which elements of a given problem space come together to form meaningful configurations through insight (Kohler, 1947; Sharps & Wertheimer, 2000; Wertheimer, 1982). Given the importance of the derivation of meaningful solution sets in most forms of complex problem solving,

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inadequate heuristic processing, which would diminish the likelihood of insightful, meaningful solutions in context, is a likely candidate for advanced students’ prob-lems with math, language arts, and the CBEST. The analytic and recognition skills involved in heuristic processing are not explicitly taught in the classroom, as are the steps to algorithmic solutions in the same domains; they are arrived at through experience with contexts and problem spaces similar to the ones of interest. Thus, the opportunities to fail at heuristic learning are great compared with the ones available for algorithmic learning and, consequently, are more likely to provide greater difficulty for students.

We emphasize that algorithm and heuristic are not euphemisms for easy and hard, respectively. There are difficult algorithms, such as the long series of steps needed to solve a complex algebraic equation, and there are simple algorithms, such as carrying the 2 when multiplying 19 × 3. There are difficult heuristics (e.g., Gamow & Cleveland, 1976), such as the problem classification involved in answer-ing the question of how much determinacy could be relied on in the computation of the orbital path of a spacecraft traveling from the Earth to the moon based on initial launch condition, in view of the fact that this is the three-body problem, which is extremely difficult to solve for large, real-world objects (e.g., Basdevant & Dalibard, 2000; the answer is that because such determinacy was relatively low, the lunar craft were built to be capable of independent rocket-powered motion to correct final trajectories.). In contrast to such complex heuristic processing, there are simple heuristics, such as “Should I eat in a restaurant where everyone I know has gotten food poisoning?” In short, heuristic and algorithmic processing are qualitatively, not merely quantitatively, different. Heuristics are implicit and require experientially based insight, whereas algorithms are explicit and perfectly transparent if the respondent has memorized the necessary steps. This distinction applies to both mathematical and natural-language processing.

However, even though heuristic processing and algorithmic processing are qualitatively different, they must be connected in some way. Recent research has provided a new theory of cognitive processing that may apply to these issues in higher education and in more general realms. The gestalt/feature-intensive process-ing (G/FI) theory (Sharps, 2003; Sharps & Nunes, 2002) suggests that in complex cognitive situations some effective heuristics are built up through feature-intensive analysis of relevant algorithmic situations until, with experience, an individual is able to use heuristic processing effectively and in a versatile manner across the cognitive domain to which the heuristics in question pertain. For example, a stu-dent in physical sciences would first learn to solve relatively simple equations and quantities in specific situations and then to handle increasingly complex series of equations with a greater scope of possible application. As a more specific example, the student may learn the basic gravitational algorithm F = GMm/d2 and gradually learn to apply it in more complex, abstract situations until he or she could recog-nize the three-body problem as a gravitational problem for which the determinacy of the basic algorithmic equation requires more creative, implicit, and, hence,

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heuristic solutions (e.g., rockets and independent pilot control capabilities on the lunar module).

With practice, the individual encodes regularities among both types of equa-tions and their uses in a variety of areas with increasing ease and celerity. The individual acquires gestalt-habit patterns of dealing with relevant equations, as defined previously and with reference to the problems to which they are to be applied, as general rules of thumb or heuristics. As this process continues, a general gestalt pattern of problem classification will be followed, resulting in hierarchical, refined, stepwise progressions toward solutions in the now more expert student (e.g., Sharps, 2003; see also Sharps & Wertheimer, 2000; Wertheimer, 1982.)

Is there evidence, however, linking heuristic processing with classroom suc-cess? The answer is yes. Researchers have repeatedly demonstrated the importance of heuristic acquisition for classroom knowledge. Silver (1981) asked junior high school students to categorize story problems in mathematics and explain the bases for their categories. He found that students who had difficulty understanding the crucial mathematical structure of the problems and did poorly in solving them tended to use story elements to classify the problems. Students who did grasp the essential mathematics and consequently performed well on the problems tended to classify on the basis of mathematical structure. Thus, good story-problem solvers engaged a heuristic of classifying in terms of math, whereas poor story-problem solvers engaged a heuristic of classifying by the story. These relatively gestalt-heuristic systems led to correct or incorrect feature-intensive analysis of the prob-lems (Sharps, 2003; Sharps & Nunes, 2002), depending on whether the heuristic employed was appropriate.

Schoenfeld (1979) demonstrated that such heuristics can be actively taught. Upper-division students in science and mathematics learned five heuristic strategies for problem solving in mathematics and demonstrated significant improvements in their performance as a result, as suggested by the current theoretical consider-ations discussed previously (e.g., Sharps, 2003; Sharps & Nunes, 2002) by earlier Gestalt considerations (e.g., Kohler, 1947, Wertheimer, 1982; see also Sharps & Wertheimer, 2000) and other work in the dynamics of understanding and the devel-opment of expertise (e.g., Bransford & Johnson, 1972; de Groot, 1965; Haviland & Clark, 1974; Kieras, 1978; Kintsch, 1979, 1994; Sharps & Martin, 2002).

Therefore, with respect to language and mathematics skills and, probably, the skills needed for the acquisition of scientific knowledge, we suggest that a crucially important factor is domain-specific heuristic competency. We propose that heuristic competency is a critical factor in the development of the level of mathematical and verbal expertise needed to pass the CBEST and in the develop-ment of a solid knowledge of the basic academic disciplines on which advanced work must be based. A student must be able to apply appropriate heuristics to a given problem domain requiring verbal or mathematical skills. We further sug-gest that similar heuristic considerations, at least in the mathematical domain, also apply to the scientific domain.

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We investigated several specific hypotheses:

Hypothesis 1 (H1): Heuristic processing capabilities in mathematics will be posi-tively associated with better practice CBEST mathematics performance and more important for such performance than algorithmic competencies.

H2: Heuristic processing capabilities in mathematics will be positively associated with better science performance and more important for such performance than algorithmic competencies. There is a close functional relation between mathematics and many cognitive elements of science, and mathematics must typically be applied to scientific questions without algorithmic precision; the fit of a particular area of mathematics (e.g., integration in math analysis, regression in statistics, selection of trigonometric functions in surveying problems) must be developed and understood heuristically, in terms of gestalt characteristics of goodness of fit (see Kohler, 1947; Sharps, 2003).

H3: Heuristic processing capabilities in verbal reasoning will be positively associated with better PCBEST verbal performance. Also, heuristic processing will be more important for such performance than algorithmic competencies.

H4: We anticipated no crossover effects. We did not expect heuristic competency in the verbal realm to have a specific influence on mathematical performance or vice versa. We anticipated the effects to be relatively domain specific.

Method

Participants

We recruited 62 (46 women, M age = 19.35 years, SD = 1.37 years; 16 men, M age = 20.56 years, SD = 2.50 years) upper-division college students for this study. Participants received extra credit in their psychology courses. No other inducements or incentives were offered.

Materials and Procedure

The CBEST is not available for research purposes, but it was important to use similar materials because the CBEST is the most studied and best character-ized of available instruments for teacher qualification. Therefore, we tested our hypotheses using a highly face- and construct-valid commercially available prac-tice CBEST (PCBEST; Obrecht, Mundsack, Orozco, & Barbato, 2001, Model Test 2; used with permission). Typical CBEST questions are shown in the Appen-dix. We compared performance on this test with scores on our ratio of heuristic to algorithmic processing (RHA) index. This index enables the evaluation of heu-ristic and algorithmic processing capabilities for a given individual in a specific content domain and provides a score reflecting this evaluation.

We investigated heuristic versus algorithmic processing through the cre-ation of an experimental index, which we termed the RHA index. This index is

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composed of an individual’s score on a series of problems requiring heuristic processing divided by that individual’s score on an equivalent series of problems requiring only algorithmic processing. We constructed and evaluated RHAs in the two crucial domains of mathematics and reading comprehension. Although we used separate post hoc evaluations of these two types of processing to clarify and explore specific findings (especially unanticipated findings) in greater detail, the ratio system offered significant advantages over simple measurement of each type of processing performance as the primary analysis type. Separate analyses of algorithmic and heuristic performance could hypothetically show a significant relation between one type of processing but not the other and a given dependent variable. However, if both types of processing contribute to a given dependent variable (PCBEST math performance, PCBEST verbal performance, or sci-ence test performance), there is no accepted way to make a precise, meaningful comparison of the relative contribution of the two types of processing involved beyond a simple assertion of bigger or smaller contributions. This problem is exacerbated in the case of the continuous independent and dependent variables involved in the present study; a simple comparison of beta weights, for example, could be misleading, especially in view of the probable collinearity of algorith-mic and heuristic processing with more general intellectual abilities (e.g., Ker-linger & Pedhazur, 1973; Mosteller & Tukey, 1977).

In contrast, the ratio formed by the RHA technique makes it possible to compare the relative contributions of heuristic and algorithmic capabilities with a given dependent variable (PCBEST math performance, PCBEST verbal perfor-mance, or science test performance) while preserving the true metric that renders this comparison mathematically meaningful. The relative contributions of the two types of processing thus manifest themselves in a single metric, contributing both to the ease of analysis and the comparison of the present results with future work in this and similar venues.

To create the materials for RHA analysis, we constructed a series of math-ematics problems involving arithmetic, basic geometry, and basic algebra and wrote three long paragraphs of text resembling a news story, an educational essay about dinosaurs, and a passage from a novel. We rendered each mathemat-ics problem in two ways: in one version, a respondent merely had to apply the relevant algorithms, whereas in the story-problem version, a respondent had to heuristically extract those algorithms from the context of the story. For example, a problem may have simply required application of the Pythagorean theorem in its algorithmic version, but in the heuristic version may have involved the height of a structure from which an observer is separated by a specific linear distance, thus requiring the respondent to recognize heuristically the necessary applications.

We wrote analogous questions to test respondents’ reading comprehen-sion of the material presented in the three paragraphs. Half of the questions were algorithmic, requiring respondents simply to locate and retrieve explicit elements of text. The other half were heuristic, requiring respondents to

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make direct inferences and recognize implicit meanings. The former algo-rithmic type of question required the respondent to engage in the formulaic process of searching the text for a specific element; the latter heuristic types of processing required the extraction of implicit meaning through gestalt or heuristic means. We asked all respondents to complete both the PCBEST and RHA index.

In addition, in consultation with natural sciences faculty in the relevant areas at California State University, Fresno, we a developed a series of questions in the natural sciences. These were factual questions, rather than items requiring spe-cific computation of outcomes, because our intent in this study was to determine the relative influence of heuristic and algorithmic processing on individuals’ scientific knowledge, rather than on their ability to make scientific computations. Although this latter skill or set of skills is an important type of knowledge, it also involves mathematics and would therefore be contaminated, in the present design, by mathematical competencies that are themselves foci of the experiment and therefore not controllable against scientific computation per se. We used questions in physics, chemistry, biology, and geology. The full set of questions is available from the first author.

For validation purposes, we presented the mathematics and reading com-prehension materials used in the RHA index to the Dean’s Committee on CBEST of the School of Education at California State University, Fresno for outside evaluation of the face and construct validity of these materials and appropriateness of the materials for the population to be addressed (upper-division college students). The committee confirmed that these materials had appropriate face and construct validity and were appropriate for this popula-tion. Sample RHA index questions are presented in the Appendix. The full sets of RHA questions are available from the first author.

Thus, these materials present equivalent problem spaces that require either more or less heuristic processing and, reciprocally, either more or less algorith-mic evaluation for both mathematics and reading-comprehension materials. The RHA index scores generated could therefore appropriately be applied to the question of basic academic skills (and related CBEST performance) in terms of their relative reliance on heuristic or algorithmic processing and the question of science education.

We scored the mathematics and reading comprehension materials for each respondent and computed appropriate RHAs separately for mathemat-ics and reading comprehension. All respondents completed the RHA index, science knowledge questions, and PCBEST (Obrecht et al., 2001) in a single testing session under group testing arrangements typical of classroom testing and of CBEST administration. Respondents required 1–2 hr to complete the entire set of instruments; they were allowed to leave the testing facility after completing of the instruments. We counterbalanced administration orders and checked all scoring twice. We used simple multiple regressions to test our

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hypotheses. We used only complete protocols for each instrument and set of questions in these analyses.

Results

The results for mathematics and science were generally consistent with the predictions of G/FI theory and our hypotheses, although the results for verbal performance were unexpected. We describe the results separately for each con-tent area.

Mathematics Skills as Measured by the PCBEST

To test the relation between mathematics RHA and mathematics skills as measured by the PCBEST, we conducted a regression analysis with math RHA as the predictor variable and PCBEST math score as the dependent vari-able. The effect of math RHA was significant on PCBEST math scores, R2 = 0.3521, F(2, 49) = 6.906, p = .0114, β = .3537, indicating the predicted relative importance of heuristic over algorithmic mathematical skills for mathematics such as those tested by the CBEST. In a second regression that evaluated ver-bal RHA against PCBEST math performance, verbal RHA was not significant for PCBEST math performance. The heuristic or algorithmic components of verbal RHA were also nonsignificant when tested separately in an exploratory regression analysis.

Verbal Skills as Measured by the PCBEST

We evaluated the relation of verbal RHA with verbal skills as measured by the verbal skills component of the PCBEST by means of a regression analysis. The effect of verbal RHA was not significant on PCBEST verbal performance. In view of this unexpected finding, we conducted an additional post hoc explor-atory analysis. We regressed the separate components of the verbal RHA, verbal heuristic score, and verbal algorithmic score against PCBEST verbal perfor-mance, which led to significant results. The results of this analysis must be inter-preted with caution. However, these exploratory results suggest that the relation between verbal heuristic score and PCBEST verbal performance was significant, R2 = 0.4856, F(4, 50) = 4.256, p = .0443, β = .2385, as was the relation between verbal algorithmic score and PCBEST verbal performance, R2 = 0.4856, F(4, 50) = 15.106, p = .0003, β = .4790. In short, heuristic processing, as hypothesized, was important for verbal performance. However, either heuristic processing did not exceed algorithmic processing in importance in this regard or its relative importance was insufficiently great to observe statistically.

The regression analysis further showed that, as expected, neither component of math RHA was significant for PCBEST verbal performance. These results

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indicate that although both heuristic and algorithmic abilities are important for verbal performance on the PCBEST, neither type of ability is statistically defin-able as more important than the other for overall verbal competency.

Science Knowledge

We evaluated the relation between mathematics RHA and science knowl-edge performance as measured by the science test in another regression analysis. The effect of mathematics RHA on science knowledge performance was signifi-cant, R2 = 0.2029, F(2, 48) = 11.805, p = .001, β = .4437. However, we found no significant effect of verbal RHA.

Discussion

In the present study, we addressed the relative influence of domain-specific heuristic and algorithmic competencies in mathematical and verbal abilities as measured by CBEST-type academic skills evaluations and in factual science knowledge across fields. Our results were consistent with the hypothesis that heuristic processing is a statistically significant predictor of performance and, further, were generally consistent with the predictions of G/FI theory, even though the pattern of results clearly differed across domains.

In mathematics, at least for these question types that are typical of the CBEST, it is clear from the significant relation of math RHA to PCBEST math performance that heuristic skills are important for college-level mathematics competency. This is in line with the predictions of G/FI theory and earlier work in mathematics education at lower educational levels (e.g., Schoenfeld, 1979; Silver, 1981).

It is possible that these results are relatively specific to the types of prob-lems used, and it is important not to generalize these results to domains outside these kinds of problems without further empirical work. The only problem types addressed here are those typical of the CBEST. Even in this domain, there are numerous other possible influences that may enhance or diminish performance. However, statistically strong relations exist between RHA performance and PCBEST performance, and these relations must be explained. The most parsimo-nious explanation of the findings obtained is that mastery of the heuristic skills of mathematics allows the student to direct the specific feature-intensive operations necessary to solve the problems in the most efficient manner. The implication of this result is obvious: Assuming that questions of the CBEST type address college-level mathematics knowledge, it is necessary to incorporate and improve heuristic training in mathematics, perhaps by techniques similar to those of Schoenfeld (1979), at all levels of mathematics education from elementary to collegiate.

These implications assume special significance when considered in light of the findings of this study that are pertinent to science knowledge. Math RHAs

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predicted individuals’ level of science knowledge and PCBEST mathematics competency. We note again that the science questions used in the present study required no computation; apparently, heuristic abilities in mathematics are related to general, noncomputational science knowledge. We can suggest an explanation for this finding, although alternative explanations are possible and further work is needed on this issue. Heuristic processing makes it possible to recognize relatively gestalt relation and frameworks among different classes of events (e.g., the ability to use relatively gestalt heuristic knowledge to organize the feature intensive ele-ments of a given problem space in mathematics). In short, a student who under-stands the framework and general nature of a mathematics problem is in a better position to create a plan to solve that problem in a stepwise-feature-intensive manner and carry out that plan systematically. We suggest that individuals may use the same types of processing to organize the relatively systematic semantic networks of science knowledge (see Bransford & Johnson, 1972), providing the appropriate frameworks in which to store, retrieve, and work with elements of scientific information. In other words, the same types of heuristic skills that may be used to organize the elements of a mathematical problem space may also be used to organize the elements of scientific schema, resulting in better memory and comprehension. If so, training in heuristic skills in formal educational settings must assume an even greater priority.

Although both algorithmic and heuristic skills were important for demon-strated verbal competencies as measured by the PCBEST, the verbal RHA was not significant for PCBEST verbal performance. This indicates that, contrary to the results obtained for mathematics and science knowledge, neither algorithmic nor heuristic skills are more important for reading comprehension. Again, there are numerous possible explanations for the absence of a significant association, and further study is needed to evaluate the spectrum of possibilities. However, we suggest a relatively parsimonious explanation. Heuristics in the mathemati-cal realm are relatively easy to distinguish from algorithms (e.g., the heuristic “focus on the equation rather than the story in a given story problem” can be easily distinguished in content from “divide the numerator by the denominator”). However, in verbal processing and reading comprehension, it may not be as easy for the reader to make this distinction. The algorithm for answering a direct question in a given piece of text may be expressed verbally as something along the following lines: “Read the text line-by-line while retaining the question to be answered in working memory. When you encounter an answer that precisely matches the requirements of the question, stop reading and report the answer.” However, the precision of the match may prove problematic here. Questions of a more heuristic type, requiring the respondent to make inferences, clearly do not fit the formal definition of an algorithm (e.g., Berlinski, 2000). Yet, the reader must still make some precision of match between a given solution provided by the text and the more gestalt, less feature-intensive question characteristic of text processing in which inference is required.

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Therefore, we suggest that there may be an additional processing continuum in need of research attention. It is well known that there are a number of qualita-tively different types of heuristics used in reasoning, such as the representative-ness heuristic, availability heuristic, simulation heuristic, and so forth. We suggest, however, that in some areas, including text processing, heuristic and algorithmic processing may be viewed on a continuum from the relatively inferential to the relatively noninferential. Heuristic processing in reading comprehension is much less structured than in mathematics. One must draw on a relatively vague and amorphous consideration of knowledge base, prior experience, and structure and semantics of the given text to infer meaning in reading comprehension, and the mechanisms involved in comprehension are not well understood. However, researchers know that people achieve insights and comprehension based on infer-ential processing and that such processing is not readily conceptualized in terms of subgoal structure, as is required for a purely algorithmic process. Based on these considerations, the absence of a significant RHA effect on PCBEST verbal competency may have resulted from a literal blurring of the RHA’s numerator with its denominator or of the type of processing termed heuristic with the type termed algorithmic in the realm of text processing.

Further research is necessary to evaluate this idea. However, if this hypoth-esis is accurate, the present research suggests that in text processing the lines between algorithmic and heuristic processing may be blurred by comparison with other realms, and that a potentially profitable specific focus may lie in the degree to which processing must be inferential. More inferential text processing would occur when subgoal structures were less evident and more difficult to construct or comprehend. This means that the difference between inferential and noninferential heuristic processing is less a question of discrete types and more a question of the ends of a continuum. If researchers show this to be the case in future work, then it is possible that one anchoring end of this continuum may lie in true algorithmic processing. The continuum may then progress through rela-tively algorithm-like, noninferential heuristic processing and ultimately through more inferential processing of the type that appears to have emerged from the processing distinction observed in the present study. Whether the anchoring end of this hypothetical continuum opposite the true algorithm lies in the insight experience investigated by Kohler (1947) is a potentially crucial question that awaits further empirical inquiry.

For the present, our results indicate the importance of heuristic abilities in higher educational processes in mathematics and science and may therefore be of specific use in diagnosing the difficulties many individuals have with academic materials such as those addressed by the CBEST. Our findings also suggest fur-ther directions of inquiry in the study of text processing that may have relevance not only for educational goals but also for understanding the nature of algorithmic and heuristic processing and inference. Our findings indicate the utility of G/FI theory in educational applications and the heuristic value for cognitive science

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of such empirical excursions into applied realms, such as educational cognitive psychology. Finally, our findings indicate that a parsimonious explanation for the difficulties students experience with the mathematical component of the CBEST may lie in difficulties with the heuristic processing of mathematical information.

AUTHOR NOTES

Matthew J. Sharps is a professor of psychology at California State University, Fresno. His current research addresses representation theory, forensic applications of cog-nitive science (including eyewitness identification and law enforcement training), applica-tion of cognition to educational and environmental educational issues, and evolution of cognition. Adam B. Hess is an instructor of criminology at California State University, Fresno, and is completing his PhD at Alliant International University, Fresno. His current research addresses the application of cognitive science to eyewitness identification and law enforcement training, cognitive approaches to educational issues, and clinical and cognitive approaches to substance abuse. Jana L. Price-Sharps is an assistant professor of psychology at Alliant International University, Fresno, and Chief Psychologist for the Fresno Police Department. Her current research addresses stress and coping, police psy-chology, psychoeducation, and substance abuse. Jane Teh completed her undergraduate education in psychology at California State University, Fresno.

REFERENCES

Ahlquist, R., Berlak, A., Castaneda, L. V., Lea, V., & Montano, T. (2004, May). Standard-ized tests, government regulations suffocating public education curriculum. California Faculty, 14–17.

Basdevant, J.-L., & Dalibard, J. (2000). The quantum mechanics solver: How to apply quantum theory to modern physics. Berlin, Germany: Springer.

Bazerman, M. H. (1998). Judgment in managerial decision making (4th ed.). New York: Wiley.Berlinski, D. (2000). The advent of the algorithm. Orlando, FL: Harcourt.Blanchard-Fields, F. (1986). Reasoning in adolescence and adults on social dilemmas

varying in emotional saliency: An adult developmental perspective. Psychology and Aging, 1, 325–333.

Bourne, L. E., Dominowski, R. L., & Loftus, E. F. (1979). Cognitive processes. New York: Prentice-Hall.

Bransford, J. D., & Johnson, M. K. (1972). Contextual prerequisites for understanding: Some investigations of comprehension and recall. Journal of Verbal Learning and Ver-bal Behavior, 11, 717–726.

Camerer, C. F. (1995). Individual decision making. In J. H. Kagel & A. E. Roth (Eds.), The handbook of experimental economics (pp. 587–703). Princeton, NJ: Princeton University Press.

Cialdini, R. B. (1988). Influence: Science and practice (2nd ed.). Glenview, IL: Scott Foresman.

de Groot, A. D. (1965). Thought and choice in chess. The Hague, Netherlands: Mouton.Denney, N. W. (1989). Everyday problem solving: Methodological issues, research findings,

and a model. In L. W. Poon, D. C. Rubin, & B. Wilson (Eds.), Everyday cognition in adult-hood and late life (pp. 330–351). Cambridge, England: Cambridge University Press.

Denney, N. W. (1990). Adult age differences in traditional and practical problem solv-ing. In E. A. Lovelace (Ed.), Aging and cognition: Mental process, self awareness and interventions (pp. 329–349). Amsterdam: Elsevier.

Dow

nloa

ded

by [

Fond

ren

Lib

rary

, Ric

e U

nive

rsity

] a

t 21:

23 1

9 N

ovem

ber

2014

84 The Journal of Psychology

Freedman, J. L., & Fraser, S. C. (1966). Compliance without pressure: The foot-in-the-door technique. Journal of Personality and Social Psychology, 4, 195–202.

Gamow, G., & Cleveland, J. M. (1976). Physics: Foundations and frontiers. Englewood Cliffs, NJ: Prentice-Hall.

Garnham, A., & Oakhill, J. (1994). Thinking and reasoning. Oxford, England: Blackwell.

Gauvain, M. (1993). The development of spatial thinking in everyday activity. Develop-mental Review, 13, 92–121.

Gilovich, T. (1992). How we know what isn’t so. New York: Free Press.Gregory, R. (Ed.). (1987). Oxford companion to the mind. New York: Oxford.Grisso, T. (1986). Evaluating competencies: Forensic assessments and instruments. New

York: Plenum.Haviland, S. E., & Clark, H. H. (1974). What’s new? Acquiring new information as a pro-

cess of comprehension. Journal of Verbal Learning and Verbal Behavior, 13, 512–521.Kahneman, D., & Tversky, A. (1972). Subjective probability: A judgement of representa-

tiveness. Cognitive Psychology, 3, 430–454.Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision making

under risk. Econometrica, 47, 263–291.Keleman, M., & Koski, W. S. (1998, April). Political symbolism, organizational entrench-

ment and the short history of the California Basic Educational Skills Test. Paper pre-sented at the meeting of the American Educational Research Association, San Diego, California.

Kerlinger, F. N., & Pedhazur, E. J. (1973). Multiple regression in behavioral research. New York: Holt, Rinehart, & Winston.

Kieras, D. E. (1978). Good and bad structure in simple paragraphs: Effects on apparent theme, reading time, and recall. Journal of Verbal Learning and Verbal Behavior, 17, 13–28.

Kintsch, W. (1979). On modeling comprehension. Educational Psychologist, 14, 3–14.Kintsch, W. (1994). Text comprehension, memory, and learning. American Psychologist,

49, 294–303.Kohler, W. (1947). Gestalt psychology. New York: Mentor.Lawton, M. P. (1982). Competence, environmental press, and adaptation of older people.

In M. P. Lawton, P. Windley, & T. Byerts (Eds.), Aging and the environment: Theoreti-cal approaches (pp. 33–59). New York: Springer.

Matlin, M. W. (2005). Cognition. New York: Wiley.Medin, D. L., & Bazerman, M. H. (1999). Broadening behavioral decision research: Mul-

tiple levels of cognitive processing. Psychonomic Bulletin & Review, 6, 533–546.Mosteller, F., & Tukey, J. W. (1977). Data analysis and regression. Reading, MA: Addi-

son-Wesley.Newell, A., & Simon, H. A. (1972). Human problem solving. Englewood Cliffs, NJ:

Prentice-Hall.Obrecht, F., Mundsack, A., Orozco, J., & Barbato, L. (2001). Barron’s how to prepare for

the CBEST (5th ed.). New York: Barron’s Educational Series.Park, D. C. (1992). Applied cognitive aging research. In F. I. M. Craik & T. A. Salthouse

(Eds.), Handbook of cognition and aging (pp. 449–493). Hillsdale, NJ: Erlbaum.Payne, J. W. (1973). Alternative approaches to decision making under risk: Moments

versus risk dimensions. Psychological Bulletin, 80, 439–453.Payne, J. W. (1982). Contingent decision behavior. Psychological Bulletin, 92, 382–402.Reese, H. W., & Rodeheaver, D. (1985). Problem solving and complex decision making.

In J. E. Birren & K. W. Schaie (Eds.), Handbook of the psychology of aging (2nd ed., pp. 474–499). New York: Van Nostrand Reinhold.

Dow

nloa

ded

by [

Fond

ren

Lib

rary

, Ric

e U

nive

rsity

] a

t 21:

23 1

9 N

ovem

ber

2014

Sharps, Hess, Price-Sharps, & Teh 85

Schoenfeld, A. H. (1979). Explicit heuristic training as a variable in problem-solving per-formance. Journal for Research in Mathematics Education, 10, 173–187.

Silver, E. A. (1981). Recall of mathematical problem information: Solving related prob-lems. Journal for Research in Mathematics Education, 12, 54–64.

Simon, H. A. (1957). Models of man. New York: Wiley.Sharps, M. J. (2003). Aging, representation, and thought: Gestalt and feature-intensive

processing. Piscataway, NJ: Transaction.Sharps, M. J., & Martin, S. S. (2002). “Mindless” decision making as a failure of contex-

tual reasoning. The Journal of Psychology 136, 272–282.Sharps, M. J., & Nunes, M. A. (2002). Gestalt and feature-intensive processing: Toward a

unified theory of human information processing. Current Psychology, 21, 68–84.Sharps, M. J., & Wertheimer, M. (2000). Gestalt perspectives on cognitive science and

experimental psychology. Review of General Psychology, 4, 315–336.Teitelbaum, P. (2003). The influence of high-school graduation requirement policies in

mathematics and science on student course-taking patterns and achievement. Educa-tional Evaluation and Policy Analysis, 25, 31–57.

Tversky, A., & Kahneman, D. (1972). Subjective probability: A judgement of representa-tiveness. Cognitive Psychology, 3, 430–454.

Tversky, A., & Kahneman, D. (1973). Availability: A heuristic for judging frequency and probability. Cognitive Psychology, 5, 207–32.

Tversky, A., & Kahneman, D. (1974). Judgement under uncertainty: Heuristics and biases. Science, 125, 1124–1131.

Wertheimer, M. (1982). Productive thinking (rev. ed.). Chicago: University of Chicago Press.

Willis, S. L. (1991). Cognition and everyday competence. Annual Review of Gerontology and Geriatrics, 11, 80–109.

Willis, S. L. (1995). Everyday problem solving in the cognitive challenged elderly. In M. Smyer, M. B. Kapp, & K. W. Schaie (Eds.), The impact of the law on aging (pp. 87–126). New York: Springer.

Willis, S. L. (1996). Everyday problem solving. In J. E. Birren & K. W. Schaie (Eds.), Handbook of the psychology of aging (4th ed., pp. 287–307). New York: Academic Press.

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APPENDIXSample Items From Instruments Used in This Study

California Basic Educational Skills Test (CBEST)Mathematics

1. 1/2 + 2/6 + 3/4 = ?

a. 1 7/12 b. 1 9/12 c. 2 7/12 d. 3

2. There were 18,000 people who came to the school banquet last year. If this year’s attendance had a 20% increase in the number of people in relation to last year, how many people came to the school banquet this year?

a. 2,160 b. 21,600 c. 3,600 d. 14,400

Reading

Questions refer to the following passage:

Many people believe that raccoons wash all of their food before they eat it. This is often the case. Raccoons often wash their food. There have even been documented cases where a raccoon refused to take food when he couldn’t find any water!

1. The next sentence in this essay is most likely:

a. The year-round home, or den, where the young are born is usually in the hollow limb or tree trunk.b. But on the other hand, raccoons have been known to eat food even when they were some distance from water, though perhaps they weren’t too happy about it.c. Raccoons give birth to young about once a year, with four or five to a litter.d. Their eyes are covered with a black mask.e. Raccoons live in places where there is water and trees for dens.

2. The best title for this essay is:

a. The Natural World and the Animals that Live Thereb. Raccoons c. The Home of the Raccoond. How to Describe a Raccoone. Raccoons Sometimes Wash Their Food

(appendix continues)

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APPENDIX (continued)

Ratio of Heuristic to Algorithmic Processing (RHA) IndexMathematics (Algorithmic)

1. S = B × P. If S = 5000 and B = 20, what is P?

A. 5000 B. 100,000 C. 20000 D. 5020 E. 250

2. If you take 16 from 200, then add 32, then divide by 2, you get A. Then multiply 5 by 27 to get B. What is A × B?

A. 108 B. 135 C. 14,580 D. 10,800 E. 24,275

Mathematics (Heuristic)

1. 15 cars, numbered 1–15, are racing in the Indianapolis 500. What is the probability that a car with a number divisible by 3 will win?

A. 1/15 B. 1/3 C. 3/15 D. 1/5 E. 1

2. The ratio of the carnivores to herbivores in Dinosaur Park is 7 to 27. If there are 567 herbivorous dinosaurs in the park, how many carnivores are there?

A. 7 B. 134 C. 34 D. 567 E. 147

Reading Comprehension

Questions refer to the following passage:

Some of the oddest and most interesting dinosaurs were the pachycephalosaurids, or bone-headed dinosaurs. The largest of these dinosaurs was the Pachycepha-losaurus itself, a creature some fifteen feet long. The skull of this animal rose into an extraordinary dome, ten inches high, on the top of the head. The dome was quite dense and solid, providing considerable protection to the brain. The dome could not have been used to help in gathering food and could not have been a practical weapon to fight predators. Paleontologists therefore hypothesize that male pachycephalosaurids may have engaged in ritual head-butting battles, presumably for mating or dominance purposes. Similar ritual battles are found among some mammals today, including bighorn sheep. Pachycephalosaurids lived during the Cretaceous period.

Reading Comprehension (Algorithmic)

1. Pachycephalosaurids were:

(appendix continues)

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APPENDIX (continued)

A. bone-headed dinosaurs. B. the largest of the dinosaurs.C. very ordinary, typical dinosaurs. D. not dinosaurs at all but another type of mammal. E. biologically related to bighorn sheep.

2. The skull of the pachycephalosaurus was: A. horned. D. fragile. B. adapted for gathering food. E. tightly packed.C. domed.

Reading Comprehension (Heuristic)

1. The author’s attitude toward these dinosaurs is one of:

A. interest. D. contempt.B. insouciance. E. affection.C. dislike.

2. Why are bighorn sheep brought into the paragraph?

A. To show the taxonomic relationship between dinosaurs and mammals.B. To show the relationship between pachycephalosaurus and bighorns.C. To show that at least some kinds of modern animals have the behavior suggested for these dinosaurs.D. To show that eventually even mammals will become extinct.E. It should not have been included; there is no reason to want to know about them in connection with this paragraph.

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Original manuscript received September 25, 2006Final version accepted May 10, 2007

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