Effects of inductive multimedia programs in mediating word problem translation misconceptions.

This study examined the effects of two inductive multimedia programs, one including the teaching and using of the coordinate graph and its language, on university students' ability conceptualize variables and create equations from word problems. The programs were designed to address the problem of syntactic and semantic translation misconceptions in world problems. For both treatments, posttest scores were significantly higher on both function construction and variable conceptualization than on the pretest. However, students receiving instructions via the inductive multimedia with graph program scored significantly higher on function construction than did those receiving the multimedia only program. This result is consistent with propositions recognizing the conceptual richness of visuals, specifically the coordinate graph, in mathematics education learning. Results suggest using inductive multimedia program treatments that incorporate many instructional strategies including inquiry learning from data, tutorial, schema, and core representational systems. This study also suggests using inductive multimedia programs that include the coordinate graph teaching strategy for the problem of translation, specifically creation of linear function.

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In today's work environment, the ability to think critically, to communicate basic mathematical ideas, and to develop problem-solving strategies is essential (Smith, 1994). Furthermore, for students building toward a career, research indicates that a strong relationship exists between mathematical skills and success in college, regardless of major (Waits & Demana, 1988). However, despite the proven short- and long-term value of such skills, student underpreparedness in mathematics is a continuing and growing problem in higher education (Berenson, Best, Stiff, & Wasik, 1990).

In algebraic problem-solving situations, students find algebraic applications difficult. Most students cannot, in fact, translate rational word problems into simple linear functions (Clement, 1982; Lewis & Mayer, 1987; Lochhead & Mestre, 1988; Mayer, 1982; Wollman, 1983). Further, students simply have difficulty with the concept of variables (Leinhardt et al., 1990; Philip, 1992; Usiskin, 1988); variables are an integral part of the concept of function.

Misconceptions are a typical source of errors. Students maintain are incorrect conceptual systems regarding specific concepts in mathematics. These misconceptions may or may not have been intentionally instructed and have been misunderstood repeatedly by learners (Leinhardt et al., 1990). A considerable amount of research has been done in the area of algebraic misconceptions (e.g., Carlson, 1998; Kaur & Boey Peng, 1994), specifically misconceptions during translation tasks from problem situations to equations (Clement, 1982; Kaput & Sims-Knight, 1983; Lewis & Mayer, 1987; Mayer, 1982; Rosnick & Clement, 1980). Generally speaking, misconceptions, evidenced by persistent error patterns, have a psychological base. According to Piagetian learning theory, learners use their existing cognitive structures and construct new knowledge that will be adapted. The problem is that "learner's existing cognitive structures are difficult to change significantly" (Herscovics, 1989, p. 62).

Misconceptions can also be viewed via current cognitive psychology. Mayer (1992) explained the matured (the 1970s and 1980s) cognitive learning theory as a three-part process: "selecting, organizing, and integrating." According to Mayer, integrating is "connecting the organized [new] information to other familiar knowledge structures already in memory" (1984, p. 33). The problem again may be that learners have difficulty opposing their already-existing cognitive structures during the connecting process, especially if their prior knowledge structures make sense to them. The following paragraphs identify two misconceptions that occur consistently during translation from problem situations to equations.

A syntactic type of misconception refers to the student's use of the syntax, or structure, of a relational word problem (or statement) to translate literally into an algebraic expression. For example, in the professor-student problem, six times the number of students is literally translated into 6S. Students sequentially replace key English words of a relational word problem with mathematical symbols from left to right (Clement, 1982; Herscovics, 1989; Lochhead & Mestre, 1988). This erroneous process is also called syntactic-translation, syntactic error, and word-order-match.

A semantic type of misconception refers to the student's consistent incorrect interpretation of relational statements as static comparisons. In the professor-student problem, for example, students use the symbol "S" to represent the group of students, the symbol "P" to represent the group of professors, and the equivalence symbol "=," not to represent a mathematical relationship but simply to separate the two groups (Clement, 1982; Herscovics, 1989; Kaput & Sims-Knight, 1983; Lochhead & Mestre, 1988). This approach is also called static-depictive, static comparison, or semantic translation.

This lack of understanding of mathematics, in algebra for example, forces students to memorize algebraic rules and procedures. Therefore, many students think algebra is simply rule-based memorization (Brown, Carpenter, Kouba, Lindquist, Silver, & Swafford, 1988; Kieran, 1992). As a result, students are often unable to apply basic algebraic and geometric concepts to problem solving (Brown et al.).

Instruction in generating linear function needs an approach that considers students' general reasoning processes (English & Warren, 1995), accounts for conceptual errors in problem translation (Kaput & Sims-Knight, 1983), and includes linked multimedia presentations (Kaput, 1992b, Mayer & Simes, 1994). Several researchers (Bishop, 1989; Clements, 1982; Janvier; 1987) have encouraged research in the exploration, investigation, and curriculum implications of graphic and tabular representations of knowledge. According to Dugdale, Thompson, Harvey, Demana, Waits, Kieran, McConnell, and Christmas (1995), a graphic representation "can reveal insights into the problem situation that are not readily revealed, by symbol manipulation alone" (p. 330).

On the other hand, graphical presentation adds its own ambiguity to the learner's syntactic translation problem (Carpenter, Corbit, Kepner, Lindquist, & Reys, 1981; Goldenberg, Lewis, & O'Keefe, 1992; Kerslake, cited in Herscovics, 1989; Monk, 1992). Despite the ambiguous role of graphic presentations in problem solving, support exists for the use of visuals: as mnemonic tools (Atkinson, 1975; Lewis, 1989; Winn, Tian-Zhu, & Schill, 1991), as gestalt-producing mental processors (Skemp, 1987), and as mathematics language (Esty, 1992; Janvier, 1987; Kaput, 1989). Graphs, for example, enable problem solving because a graph allows one to view a single graphic entity instead of a binary quantitative relationship (Kaput, 1989). Kosslyn (1994) has shown that human visual perception and cognition have strengths and limitations, and their measures depend on both the quality of visuals displayed and the adequate usage of those visuals.

Clements (1982), a recognized expert in mathematics education, has concluded that despite the fact that clear guidelines for the use of visuals in classroom practice have not yet emerged, "there should not be a reduction in the amount of research which is aimed at achieving this end" (p. 36). Clements stressed that "Mathematics educators need to develop better instruments for assessing the role of visual imagery in mathematics learning" (p. 36). The classroom applications of computer-supported or computer-generated graphs and of graphic language are important and viable areas of research in mathematics teaching (Bell & Janvier, 1981; Bishop, 1989; Eisenberg & Dreyfus, 1991; Lesh, 1987).

To address the difficulty students have in learning how to generate linear functions (or equations) from relational word problems, mathematic education literature offers a two-part solution. Using linked representational media (multimedia programs), mathematics Computer-Based Instruction (CBI) should stress (1) inductive problem-solving strategies or scientific heuristics (for example, by working backwards, working inductively, or applying algebraic thinking to data) (Martinello & Cook, 2000, DeMarois, McGowen, & Whitkanack, 1996; Polya, 1954a, 1954b) and, most importantly, (2) the teaching of the language of mathematics and math visuals (graphs) (Esty, 1992; Kaput, 1992b; Bell & Janvier, 1981).

The present study considered a combined approach regarding the role of visuals in problem solving. This approach is based on the need that exists for inductive algebraic multimedia software programs that strongly support the growing interest in the use of visuals in mathematics-based cognitive processes. The graphic presentation and construction capabilities of software programming currently offer the most practical way of producing and using quality visuals. The mere inclusion of graphs is not enough, however; programs should also address the language of graphs and provide solutions to students' graphic misconceptions in a dialogue with learners. Algebraic software programs should consider graphic cognitive obstacles, graph language, and the order of visual presentations. The present study specifically looked for a strong possible visual effect for the combination of graphs with graph language in an inductive multimedia program to improve linear function construction and conceptualization of variables.

The purpose of this study was to determine the effects of two inductive multimedia programs, one including graphs and graph language, on subjects' ability to create linear functions and construct variables from word problems. There were two primary hypotheses for this study: First, students receiving instructions via either software program would score higher on the posttest than on the pretest in both areas: linear function construction and variable conceptualization. Second, students receiving instructions via the inductive graph program would score higher on the posttest in both areas, linear function (equation) construction and variable conceptualization, than would students receiving the multimedia treatment only.

Method

Subjects

The subjects were 98 undergraduate students enrolled in a computer literacy course offered by the College of Education of a large southwestern university during the fall 1997 semester. This course focuses on computer applications and their uses in education and business. Although this course is not required of all students, many students view it as a requirement and enroll in this course rather than taking an additional math course to fulfill part of the university's numeracy requirement. The majority of subjects who enroll in this course are freshmen majoring in the social sciences, including education. Subjects who volunteered for the study received extra credit for participating and were randomly assigned to one of two groups. Subjects ranged in age from 18 to 25 years old.

Materials

Program Contents

Two inductive multimedia programs served as the instructional treatments for this study. Both self-paced treatments were developed by the author using

Authorware (Authorware, Inc., 1989) via Macintosh. Both programs, InductiveThinker Table and InductiveThinker Table & Graph, had two lessons. Lesson One contained information about the input, output, and independent and dependent variables. Lesson Two included information about the rate of change (or the slope of the function) and linear function creation. The pace was user controlled and subjects had the ability to navigate between pages, sections, or lessons and could exit the program at any time.

The first program, InductiveThinker Table, included tables in addition to other elements (e.g., animation). A series of screens addresses the misconceptions or syntax errors, while using a method to help users understand these misconceptions. The program addresses the translation problem, syntactic-translation, in two steps: lesson one discusses the misconceptions about "Variables" that are treated as nouns, etc., and lesson two addresses the "Slope," in our case the "Constant Rate of Change," treated as an adjective modifier instead of multiplicative coefficient. The concept of variable is introduced in lesson one and the concept of slope is introduced in lesson two.

The program starts with a navigation page and a two-page overview to clarify the directions of the program. A menu page directs students in selecting a lesson to explore. The sequence of lessons is controlled by the program. Learners must go through Lesson One prior to Lesson Two.

InductiveThinker Table uses the table of value, suggested by many experts, in both lessons to address the translation problem. In Lesson One, learners are introduced to the concept of data, input-output, and independent-dependent variables. Learners view a real rational word problem via a video segment and then are asked to help organize a table translating the word problem into an organized data using language that is familiar--input-output. Eventually the concept of variable is presented along with the typical misconceptions. Lesson Two reviews the concept of variables and addresses the concept of rate, again identifying misconceptions to the learners.

The software not only uses a table of values to address the concept of rate, it uses the technique of patterning within the table to clarify a logical reasoning embedded in the concept of rate. The software not only opens a discussion with the learners, it helps learners evaluate their learning by introducing very simple evaluation questions in the form of reflection. These examples from both lessons are themselves mini-programs. "Last note," at the end of the lesson, reminds learners of the most important goal of that lesson. Example screens are presented in Figures 1 through 6.

[FIGURES 1-6 OMITTED]

The second version of the program, InductiveThinker Table & Graph, included both tables and graphs. This program was identical to the first version in all respects except for the addition of the graph visuals and graph-related text. The Table and Graph Program teaches the language of the coordinate graph to the learners by:

* Helping them to focus on the more global or group of points of the graph instead of features of an individual point of the graph (Figure 7).

[FIGURE 7 OMITTED]

* Showing them that a line can represent a linear mathematical equation graphically (Figure 8).

[FIGURE 8 OMITTED]

* Illustrating the fact that the coordinate graph is a graphical representation of a raw data that is holistic and complete (Figure 9).

[FIGURE 9 OMITTED]

* Teaching them how to find the position of any point in a plane from the point's distances or coordinates from fixed lines called the "axes" of coordinates (Figure 10).

[FIGURE 10 OMITTED]

* Telling them the rules, using consistent units of distance, demonstrating that the axes of coordinates are perpendicular and intersect at a point called the "origin" of the coordinates.

* Making the rule that a point's coordinates and coordinate lines are specified using algebraic symbols, such as x, y, and z (Figure 11).

[FIGURE 11 OMITTED]

* Illustrating the fact that analytical geometry brings analytical (or symbolic and numeric) algebra into geometry and algebraic problems can be solved by geometric methods (Figure 12).

[FIGURE 12 OMITTED]

Each lesson in both programs provided (1) general theme(s), (2) questions, (3) answers, (4) practice, and (5) summary. Each lesson began with a sentence or two describing the theme (objective) of the lesson. A real mathematical problem was then demonstrated via ordinary, familiar language. In the "investigation" section, the program encouraged learners to guess, create, and draw a mathematical model for the proposed problem through a series of questions. The questions were answered in detail by the program in the "discussion" section. In the "practice" section, the software introduced a series of related questions with levels of feedback.

Theoretical Framework of the Programs

The programs' construction was theoretically and pedagogically based. The software programs were designed according to findings in inductive learning (Bruner, 1966; Polya, 1954a, 19954b), metacognition (Schoenfeld, 1992), syntactic misconceptions (Goldenberg, Lewis, & O'Keefe, 1992), and graphic literacy (Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). For example, InductiveThinker includes the ability for learners to write short notes in an informal reasoning environment and explains how data tables will relate to the generating theory via mathematical reasoning. InductiveThinker is constantly in an interaction mode, allowing learners to be focused and engaged in an inductive dialogue.

Throughout the program, while symbolic manipulation is performed by the computer, learners are encouraged in the creation of algebraic function from the data from sets of ordered pairs. Learners are asked, repeatedly, to conjecture in deriving these sets to avoid syntactic misconceptions, InductiveThinker provides learning from graphs as well. Learners construct graphs from their components, specifically the axis to derive the line, providing pictorial movement of an experience via presentation of its real environment. The overall goal is to describe and avoid the subject's conceptual blur and to provide alternatives (tables or table and graph) to allow learners to become aware of their cognitive strengths and weaknesses in problem solving situations.

Program Evaluations

The instructional treatments were evaluated via two types of formative evaluation procedures described by Dick and Carey (1996): one-to-one and a field test. The instructional, cosmetic, program, and curriculum adequacies of the treatments were reviewed by three professional Computer-Based Training (CBT) designers employed at major corporations in the southwest and two instructional software development instructors on the faculty of a major university. In addition, the mathematics instructional adequacy of the programs was approved by two mathematics educators in the College of Education of the university.

After the peer review (or one-to-one evaluation) which suggested a major change regarding the second version, the treatments were piloted using seven students from the target population who were unfamiliar with the material covered in the programs. The purpose of the pilot study was to test the instructional materials for the treatment and the measurement instruments. Attitudes, general comments, and suggestions from this group were used to revise the programs. For example, the program's lessons were made shorter and easier to understand.

The overall pilot pretest result for the creation of linear functions was 38% (M = 2.3 correct out of 6 function questions) and the overall posttest result was 60% (M = 3.6 correct out of 6 function creation questions), an increase of 23%. The overall pilot pretest result for the variable conceptualization was 58% (M = 3.5 correct out of 6 variable questions) and the overall posttest result was 72% (M = 4.3 correct out of 6 variable questions), an increase of 14%. The pilot questionnaire indicated students' strong positive attitudes regarding the methods of instruction, the pace of the programs, the mathematics and the computers.

Criterion Measures

Three instruments were used to collect data.

The first instrument was a 12-item pretest administered to both groups at the beginning of the data collection as one of the primary criterion measures. Test items were presented in short answer form (both English and algebraic) and were paralleled to the practice items contained in InductiveThinker Table, the first program. The pretest items measured student achievement on the instructional themes specifically stated in the InductiveThinker Table program. Examples of pretest questions include:

Item 1: There are six times as many students as there are professors at this university. Write an equation that represents this situation.

Item 4: Trees are cut and new ones are planted. The data are shown below.

Number of Trees Planted   Number of Trees Cut
          P                       C

          2                       4
          4                       8
          7                      14
          8                      16
        100                     200

a) Write an equation that will allow you to predict the number of trees planted (P) given the number of trees cut (C).

b) Write a sentence in English that provides the same information as the equation you just wrote.

The second instrument was a 12-item posttest that was administered to both groups at the end of the data collection as another of the primary criterion measures. Test items were again in short answer form (both English and algebraic) and were parallel to the items contained in the pretest and to the practice items contained in InductiveThinker Table, the first program. The first six items of both pretest and posttest were related to linear function construction. These six questions paralleled the classic student-professor example to some extent. The next six items of both pretest and posttest were related to variable construction conceptualization. Examples of posttest questions include

Item 1: There are thirty times as many students as there are classes at this university. Write an equation that represents this situation.

Item 4: A company donates money to a charity according to the donations of its employees. The data are shown below.

Dollars Donated by Employees   Dollars Donated by Company
              E                            C

             20                           60
             32                           96
             35                          105
            400                         1200

a) Write an equation that will allow you to predict the number of dollars paid by the company (C) given the number of dollars its employees donated (E).

b) Write a sentence in English that provides the same information as the equation you just wrote.

A split-half (odd-even) reliability test was conducted on the 12 items pretest and posttest to assess linear function construction and variable conceptualization. Coefficient alphas were computed to obtain internal consistency estimates of reliability for these two scales. The alphas for the pretest and posttest scales were .70 and .61 respectively. The tests measured the intended content areas: Function construction and variable conceptualization. Pretest and posttest items were parallel to the practice items contained within the instructional programs and were short-answer and multiple choice in format.

Two rubrics for grading the pretest and posttest were developed. Subjects received one point for each correct answer. Partial credit was allowed for questions 7, 8, 9, 10, and 11, which required explanation in addition to the correct answer. If subjects provided the correct answer but did not provide an explanation, they received half a point for the item.

The third instrument was a 14-item Likert-type questionnaire with one being Strongly Agree and four being Strongly Disagree. These items measured learner beliefs regarding conceptual understanding, method of instructional design, pace of the software, and mathematics via computers.

Procedures

Subjects were randomly assigned to two treatment groups. Both treatment programs were delivered via Macintosh computers in a computer lab in the College of Education. Subjects spent as much time as they desired on any part of the program. The length of the treatment was about an hour. Subjects from each group viewed the treatments individually under the same physical conditions and all received extra credit for participating in the study. Subjects chose their own participation times from a variety of times offered and were tested via a pretest before they viewed the programs and via a posttest immediately upon completion of the programs. The student questionnaire was administrated to the students immediately after the posttest. Data collection was conducted over two weeks.

Design and Data Analysis

This study used a Pretest-Posttest Control Group design, which requires a minimum of two groups formed by random assignment. A pretest of the dependent variables was administrated to both groups; the experimental group received a manipulated treatment (Inductive Table and Graph) and both groups took a posttest. The Pretest-Posttest Control Group was an experimental design because the second version of the inductive program (Inductive Table and Graph) is a manipulated version of the first version (Inductive Table) and all subjects were randomly assigned to the treatments. The treatment forms were A versus A+ B.

The Pretest-Posttest Control Group was selected because the subjects had different mathematical backgrounds and came from different disciplines. The combination of random assignment, the presence of a pretest, and the presence of a control group controlled threats to internal validity. However, the Pretest-Posttest design may produce confounding variables, in which an interaction between dependent and independent variables may occur. This confounding effect was controlled because the study's sample size was large, the subjects were randomly assigned to the treatments (which eliminates the systematic bias), and the study had a well-designed pretest-posttest instrument.

The Pretest-Posttest Control Group design of this study attempted to control for many extraneous variables affecting performance on the dependent variables. The extraneous variables included history, maturation, testing, instrumentation, regression, selection, and mortality. Maturation was not a threat to this study because neither significant time nor events elapsed during the one-to-two hours of subjects' involvement in the study. The possibility of improved scores on posttest from pretest interaction was addressed by the study's large sample size, randomly assigned treatment, and well-designed pretest-posttest instrument; in addition, both groups would have been equally subject to this confounding effect. Subjects were not selected based on extreme scores, nor were they studied via already-formed groups. The short duration of this study reduced the threat of mortality.

Analysis of Covariance (ANCOVA) or Multiple Analysis of Covariance (MANCOVA) adjusted the posttest means to what they would be if all groups started out equally on the covariate at the grand mean. The main purpose of the covariance (pretests) in ANCOVA or MANCOVA is to reduce error variance. The primary hypothesis of this study was that there would be a significant difference between the two groups, based on independent variables (two inductive software treatments) with two dependent variables (linear function creation and variable conceptualization).

Results

Hypothesis One: Overall Treatment Effects

A one-way repeated-measure ANOVA was conducted with the factor being treatment and the dependent variable being the pretest and posttest scores for function construction. The means, highest possible scores, number of subjects, and standard deviations of scores are presented in Table 1. The results for the ANOVA showed a significant treatment effect on function construction regardless of kind of effect, Wilks's Lambda = .457, F(1, 97) = 115.27, p < .001, multivariate d (Eta Squared) = .543.

A follow-up pairwise comparison, univariate test was conducted. The results confirmed the Multivariate test indicating that the posttest mean (M = 4.11, SD = 1.52) was significantly greater than the pretest mean (M 1.97, SD = 1.79), t (97) = 10.74, p < 0.008.

Another one-way, repeated-measure ANOVA was conducted with the factor being the treatment and the dependent variable this time being the pretest and posttest scores for variable conceptualization. The means, highest possible score, number of subjects, and standard deviations of scores are presented in Table 2. The results for the ANOVA showed a significant treatment effect on variable conceptualization regardless of kind of effect, Wilks's Lambda = .622, F(1, 97) = 58.84, p < .001, multivariate d = .378.

A follow-up pairwise comparison, univariate test was conducted. The results confirmed the Multivariate test indicating that the posttest mean (M = 5.01, SD = .83) was significantly greater than the pretest mean (M = 4.06, SD = 1.21), t(97) = 7.68, p < .001.

Hypothesis Two: Inductive Table versus Inductive Table and Graph

A MANCOVA with two dependent variables and two covariates was conducted. The independent variable, instructional treatment, included two levels: inductive table and inductive table and graph programs. The dependent variable, posttest scores, also included two levels: function construction and variable conceptualization. The covariates were pretest scores on both function construction and variable conceptualizations.

The MANCOVA indicated that the adjusted population mean vectors (posttest scores) were significantly different among the groups at the .05 level (F = 4.77, p < .001). The MANCOVA's first assumption (check to see that there is a significant relationship between the dependent variables and the covariates) was verified. Sample size justified using two covariates (C < 8, where C is the number of covariates).

A separate analysis of covariance was calculated on each dependent variable. The probability indicated that only the function construction variable was significant at the level .05 (Table 3). The power on the variable conceptualization was .31. The probability of students receiving instructions via either treatment and scoring significantly differently on the posttest in variable conceptualization is very low. Low power indicates a low probability of rejecting the null hypothesis.

The multivariate test for the homogeneity of the regression hyperplanes was not significant at the .05 level (F = .237, p < .917) indicating that the assumption of homogeneity was quite tenable. The multivariate F, corresponding to Wilks's Lambda, indicated that there was a significant difference between the set of dependent variables and the set of covariates at the .05 level (F = 6.367, p < .001). Table 4 indicates that 22.83% of the within variability on variable function construction is accounted for by two covariates, pretest function construction and variable conceptualization.

The Bryant-Paulson procedure was conducted because of the possibility of measurement error on the covariant of low or questionable reliability. The question was whether there was a significant difference between the adjusted means on function construction for the groups. Results again indicated a significant difference, Hotelling Value = .0001, interpolated critical value = 2.83, BP = 5.78.

Beliefs

MANCOVA indicated that there were no significant differences by treatment on the belief questionnaire items. The responses were scored within the range of 1 to 4, with 1 representing "Strongly agree" and 4 representing "Strongly disagree." Overall mean scores revealed that subjects across both treatments: would recommend the programs to other students (M = 1.65); thought the program made them think a lot (M = 1.83); understood the concept of variable (M = 1.64); liked the way the program taught the concept of variables (M = 1.69); and thought that over all the programs were good learning experiences (M = 1.55).

Discussion

Hypothesis One

Hypothesis One predicted that students receiving instructions via either software program would score higher on the posttest than on the pretest in both areas: linear function construction and variable conceptualization.

The results of the current study support Hypothesis One. Students, regardless of group or treatment, scored significantly higher on posttest than pretest on both function construction and variable conceptualization.

This improvement is not likely to be the result of learning during pretest. The pretest did not measure factual information which could be recalled. Rather, the translation problem is a cognitive obstacle. Subjects needed an effective treatment that addressed conceptual understanding. "Taking a pretest on algebraic equations ... is much less likely to improve performance on a similar posttest" (Gay, 1992, p. 304). Practice reinforces but is not likely to change translation processes. Cognitive "learning is seen as the correction of errors and the modification of existing schemata" (Phye, 1986, p. 141). Nor is the length of time since students last took a math class an important factor since even engineering students are unable to overcome the interference of natural language in the conceptualization of variable.

The results of this study are consistent with the finding of Wollman's (1983) study; Using a tutorial strategy, Wollman found that 94 percent of his subjects constructed the correct equation. However, later Wollman (1985) agreed that his subjects could have used the checking procedure to construct equations regardless of conceptual understanding. Observations conducted during the current study revealed that students did not check their answers during the tests and used either table or graphical procedures using arbitrary or given data to create their equations.

The instruments of this study measured what they were intended to measure: students' conceptualization of variables and, ultimately, their construction of linear functions. The treatments for this study were designed according to mathematics education research findings regarding inductive learning (logical reasoning), metacognition, syntactic misconceptions, and graphical literacy. The programs used an inquiry tutorial strategy as opposed to the checking procedure strategy used by Wollman's (1985) students. Results from the belief survey indicated that students believe the programs made them think. Given realistic situations, students had to write the functions by applying algebraic inquiry thinking (Dewey, 1938) to the data.

Results also indicated that all students did significantly better on the variable conceptualization portion of the posttest. Students' success with variable conceptualization may have positively impacted their success with function construction. If this is the case, then Clement's (1982) report that variable understanding is one of the key issues in successful problem solution is supported by the current study.

The results might have been influenced by the combination of other instructional strategies represented by the current study's instructional treatments. These instructional strategies include learning from (1) mathematical thinking, (2) schema training, (3) linked representational systems, and (4) the coordinate graph.

The effects of mathematical thinking required by the current study's programs might have caused the improved results of the present study. Shoenfeld's (1992) mathematical thinking is embedded in engagement in scientific research, the science of patterns, and the determination of regularities in systems. The methodological framework of the InductiveThinker treatments included these attributes. The treatments of the current study include the inductive methodology of constructivist epistemology for reasoning and discovering via the construction of tables of variable values.

There might be an interaction effect between the improved results of the current study and the schema-training nature of the treatments. The current study's treatments include schema acquisition that eventually provides rule automation and strengthens metacognitional skills. By using tables of values and describing steps of procedures, for example, treatments increase memory demand that could hinder student's rearrangement of information and ability to construct equations.

The results could also be attributed to the anchored linked-representational systems (Kaput, 1992b, Mayer & Simes, 1994) property of the treatments. The current study's treatments use an inductive tutorial mode to propose, measure, and evaluate working hypotheses by representing linked representations of mathematical notations, including table, verbal, and graph representations. The programs ask learners to write down their thinking and working models, then immediately compare what they have written with appropriate examples. Students receive tabular, graphic, and numerical feedback instantly during mindful engagement with the treatment programs. In addition to the linked-representational effect, other media characteristics, such as mind-machine collaboration, could be a positive attribute.

Hypothesis Two: Inductive Table versus Inductive Table and Graph

Hypothesis Two predicted that students receiving instructions via the inductive table-and-graph program would score higher on the posttest in both areas than would students receiving the table-only treatment. The results are mixed. The MANCOVA indicated that the adjusted posttest scores were significantly different among the groups. However, a separate analysis of covariance revealed that only the function construction variable was significant at the .05 level. The Bryant-Paulson procedure further verified this result.

The significant difference is not likely the result of the fact that Treatment 2 is a longer treatment. The inductive table-and-graph program, Treatment 2, included the coordinate graph strategy in addition to other strategies which were in Treatment 1. By teaching the language of coordinate graph, the coordinate graph strategy used its attributes, including geometrical graphic representation, to further subjects' conceptual understanding of the translational task. Students' difficulty with translational tasks results from their use of natural language syntax and their lack of conceptual understanding of variables and function construction, not from the amount of instruction received. Persistence of translational problems has been detected among freshmen engineering students (Clement, 1982) who received extended mathematics instructions. The difference is likely to be the result of the alternative and graphical core representation of the coordinate graph.

The multivariate results related to the second hypothesis of this study indicating that the graph has a significant effect in function construction are consistent with the findings of Tall and Thomas (1989), and Yerushalmy (cited in Kieran, 1992).

The most important outcome of the current study is the second finding, that students receiving instructions via the inductive table-and-graph program scored significantly higher on function construction of the posttest than did students receiving the table-only treatment. Treatment 2, InductiveThinker Table and Graph, includes all elements of the first treatment as well as graphic representation and the teaching of the language of the graph elements. The second finding is attributed to the graphic strategy of Treatment 2 and is most likely related to the teaching of the graphic language by the treatment program. Linked representations, including graphics alone, is not likely to be a major factor, because graphic mediation has its own ambiguity that adds to the learner's syntactic translation problem (Goldenberg, Lewis, & O'Keefe, 1992; Kerslake, cited in Herscovics, 1989; Monk, 1992).

The impact of Treatment 2 on translational learning resulted from teaching the language of mathematics, which is better addressed via an inquiry approach, not memorization, and which is very different from the English language. In addition, Treatment 2 taught the language of the graph, which is the visual language of mathematics. The treatment used full advantage of computer technology by providing immediate feedback while teaching the concepts and by applying the data to the construction of functions. In Treatment 1, learners could click on one variable and see its immediate effect on another variable. In Treatment 2, learners could view pairs of values in a table and their corresponding points on the coordinate graph.

Contrary to expectation, investigation of the second hypothesis revealed that the groups were not significantly different regarding variable conceptualization. One possible explanation for this finding is the length of Treatment 2, which was about half an hour longer than treatment 1. The last questions of the posttest were related to variable conceptualization. Observations during data collection revealed that most students rushed through answering the posttest questions; many came late and, after the posttest, had to leave quickly to attend classes across campus. Another explanation might be the fact that, according to a few observation notes, posttest and pretest questions were very similar and students may have thought they were the same questions. The result could also be attributed to the possible testing and pretest-treatment interaction effect. A careful reexamination of the pretest-posttest measures indicated that tables that were used in pretests and posttests for explanation of the variable questions were not used in the function construction sections. This may have caused students to score higher on the pretest on both areas. Another possible explanation is that the tests might not have been extensive enough to measure variable conceptualization.

Beliefs

The belief questionnaire data revealed that overall opinions regarding the programs were very favorable (M = 1.76). Students reported positive beliefs regarding inductive multimedia instruction for (a) conceptual understanding of mathematics: items four, six, and seven (M = 1.75), (b) method of instruction: items eight, ten, and eleven (M = 1.56), and (c) instructional delivery: items two, three, and twelve (M = 1.9). Students in the Inductive Table and Graph group responded very favorably (M = 1.87) to the statement "I can now translate word problems to algebraic notations" and strongly agreed (M = 1.57) with the statement "The program made me think a lot."

Implications of the Study

Results from the present study suggest that for the problem of translation, schools may find it most beneficial to use treatments similar to InductiveThinker Table and Graph, which employ multiple learning strategies, including inquiry learning from data, tutorial, schema, and core representational systems. The present study indicates that the coordinate graph strategy is an important representational system and is very effective in translational tasks only when its language is taught and that language is understood by learners. Otherwise, the implication is that the use of coordinate graphs simply adds to the learner's syntactic translational problem. Difficulties regarding graph integration, construction, and translation must be recognized and taught to learners before they use graphs.

Before students learn how to construct functions using any of the proposed instructional solutions to the problem of translation, they need to understand the concept and use of variables (Usiskin, 1988). This study did not find significant differences between groups in the area of variable conceptualization; subjects earned maximum scores regardless of the group to which they belonged. However, function construction includes variable recognition. Students must learn to conceptualize variables in order to understand functions.

Recommendations for Future Research

Further research should include: (a) Expanded Designs: Designs that include a third group that is not exposed to a pretest to reveal possible pretest interaction, (b) Factorial Designs: Designs that consider the joint effect (joint interactions) of the treatments, and (c) Qualitative Designs: Designs in which students' mathematical (translational) thinking is recorded and observed, and in which subjects are interviewed regarding their experience with the treatments.

Experimental research should consider other independent factors beside treatments, including gender, IQ, and mathematical background. Such research should measure the possible joint interactions of these factors on the study's output. The strength of this design would be the examination of a joint effect of gender and treatment on the dependent variable(s).

Much qualitative research (Bell & Janvier, 1981; Kaput, 1992a; Monk, 1992) has been conducted in the function conceptualization area. Further research is needed, specifically in the area of problem translation. Using the treatments of the current study, qualitative researchers could: (a) record students' responses (for example, combination of talking aloud and typing) to the program's questions during students' engagement with the programs; (b) record students' mathematics backgrounds via standard tests and student portfolios; (c) interview students regarding their conceptual understanding of variables and construction of functions; and (c) document students' attitudes toward the treatments, algebra, functions, and variables.

Regarding function conceptualization, Dreyfus and Eisenberg (1983) said, "The challenge is clear; the problem is well defined. We must teach so that our students will be able to grasp global notions and find inter-relationships." Still, a decade later, MacGregor and Stacey (1993) reported that "One of the greatest difficulties for beginners in algebra is linking a mathematical situations to its formal description." Students use a naturalistic representational system--the English language--to translate word problems to functions instead of using a variety of mathematics representational systems that includes the coordinate graph. The current study demonstrates how educators can use a combined treatment including the coordinate graph to help students overcome the difficulty or misconception of the translational tasks.

Table 1
Means, Highest Possible Score, Number of Subjects,
and Standard Deviations for Function Construction Scores

Dependent Variable    M     Highest Score   N     SD

Pretest              1.97       6.00        98   1.79
Posttest             4.11       6.00        98   1.52

Table 2
Means, Highest Score, Number of Subjects, and Standard
Deviations for Variable Conceptualization Scores

Dependent Variable    M     Highest Score   N     SD

Pretest              4.06       6.00        98   1.21
Posttest             5.01       6.00        98   0.83

Table 3
Analysis of Covariance on Dependent Variables

Variable                      MS       F       p     Power

Function Creation            16.38    8.3    .005     .81
Variable Conceptualization    1.33    2.2    .139     .31

Table 4
Univariate Tests for Relationship

Between Dependent Variables and Covariates

VARIABLE                      SS      MS       F       P

Function Creation            22.83   11.42    5.8    .004
Variable Conceptualization    8.68    4.34    7.3    .001

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