Jl. of Computers in Mathematics and Science Teaching (2014) 33(4), 409-427
Excel Spreadsheets for Algebra: Improving Mental Modeling for Problem Solving Jason Engerman, Matthew Rusek, and Roy Clariana Penn State University, USA
[email protected] [email protected] [email protected] This experiment investigates the effectiveness of Excel spreadsheets in a high school algebra class. Students in the experiment group convincingly outperformed the control group on a post lesson assessment. The student responses, teacher reflections involving Excel spreadsheet suggests that it operated as a mindtool, which formed the users’ mental models more similar to the tool itself. The engagement with the tool allowed students to easily replicate its functionality in this problem solving task opposed to the limited strategies used by the control group of students taught through worked examples. By working with Excel spreadsheets, the students were able to better identify manipulation of their problem spaces during the discovering solutions phase of problem solving. Subsequently, this development allowed for deeper levels of understanding algebraic relations and lead to high road transfer of the content.
Introduction For many students, Algebra is a new level of abstraction of number relations. It can be difficult for students to fully conceptualize numbers being replaced by variables whose relations are expressed as equations that can be “solved” and even visualized. Often, students can perform routine calculations with fluency, but they fail to understand the purpose for the pro-
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cedures they perform. For example, it has been observed that in the high school examined in this study, juniors and seniors enrolled in Advanced Placement Statistics skillfully computed the slope of a regression equation, but were unable to interpret its meaning as a rate of change in the context of a problem (Matt Rusek, Personal Communication, September 2008). We propose that because high school students lack a sufficient understanding of the relationship between symbolic, graphical, and tabular representations of functions, they are not fully able to transfer the learned mathematics skills to various contexts. One strategy for reducing the level of complexity and abstraction is by presenting the relationships graphically. Working with Excel spreadsheets allows students to better identify and manipulate problem spaces within the discovering solutions phase of problem solving. Problem spaces being the mental spaces in the student’s mind that are used to solve the problem (Jonassen, 2003). Subsequently, this development allows for high road transfer of the content. Several studies describe the usefulness of Excel spreadsheets in math classrooms from primary school through tertiary school (González-Calero, Arnau, & Puig, 2013; Lavidas, Komis, & Gialamas, 2013; Ntsohi, 2013; Tabach, Hershkowitz, & Dreyfus, 2013; Wu & Wong, 2007). According to these studies despite common difficulties in algebraic reasoning and Excel spreadsheet use, they enhance the ability of teaching and learning Algebra as these spreadsheets take over low-level computations and help establish relations between variables. These studies have focused on how Excel spreadsheet use benefits math learners in Algebra classes, and they describe problem-solving development in terms of a shift from arithmetic (paper-and-pencil) reasoning to algebraic reasoning. Although past studies on Excel spreadsheets and math attempt to explain several benefits of utilizing these tools, only a few studies explain why they are so effective. Wu and Wong (2007) looked at the impact of spreadsheet templates as teaching tools. Their study of twenty Singapore secondary school students considered how the templates would affect student understanding of statistical graphs (Wu & Wong, 2007). Their intervention involved no classroom teaching of statistical graphs, as the classes relied solely on four well-developed templates for exploration. Wu and Wong’s findings suggest positive impacts of this type of exploration on students’ understanding of statistical graphs and recommended further efforts to extend the use of spreadsheet to other statistics and math topics (2007). Furthermore, their findings support the premise that the spreadsheet templates were powerful enough to extend students’ understanding of statistical graphs independently, but they did not seek to explain why their students succeeded at such a high level.
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Building on this discussion, the current investigation proposes that utilizing spreadsheets for math instruction may establish strong mental models, as spreadsheets showed to behave as a mindtools that can help facilitate the transfer of content. Spreadsheets therefore allow for high transfer of mathematical knowledge. High road transfer, being a mechanism of far transfer, involves a deliberate and thoughtful search for connections, which leads to students’ ability to transfer knowledge into different contexts and performances (Perkins & Salomon, 1992). Our cognitive approach adds to the discussion of spreadsheets as cognitive tools and seeks to explain why students are better able to perform mathematical tasks as a result of these dynamic models. In the following sections, we will discuss the literature that helps to build the framework of our argument. Dynamic Modeling Tools and Mental Modeling Excel spreadsheets can be considered a dynamic modeling tool, as dynamic and executable modeling tools generally have the following attributes: (1) variables, rules, and mathematical equations embedded in the software can be executed; (2) the output and behavior of the model change over time; (3) changes are calculated and outputs are visually displayed; and (4) new variables can be added and relationships to existing variables defined (Clariana & Strobel, 2007). Spreadsheets fit these criteria and can therefore be considered dynamic tools. While utilizing dynamic models, students build mental models in such a way that exchanges occur between their understanding of the mathematical concept and the digital models they create (p. 331). This suggests that the two models, internal and external (mental and digital), are continuously affecting one another. In other words, as students try to make sense of the content, their interaction with the external dynamic models will alter their mental constructs for understanding. Mental representations play a critical role in understanding content. Hiebert and Carpenter (1992) defined mathematical understanding as the building up of a conceptual context or structure by way of mental representations through a network of representations. According to Qin and Simon (1992), mental modeling is an essential component to mathematical understanding. Qin and Simon recall that in their study of the importance of mental models in math, participants did not achieve an understanding of the equations without using mental models for representation (Qin & Simon, 1992). In fact, their conclusion contended that the reading and understanding process required the participants to alter their mental models in long-
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term memory, as these models contain the knowledge needed to form the new representations required in problem-solving (Qin & Simon, 1992). In other words, one must form mental constructs of the content before tone can act upon them. Furthermore, students begin with a naïve generic computational mental model; exploring functions in Excel then alters their model to a domain-specific (algebraic) mental model. From Solving Problems to Problem Solving In algebra class we try to develop generic skills for solving problems. Math educators would admit that their work is based on creating environments for students to become effective at solving algebra problems. A common model used by algebra teachers is known as the worked-example effect, where a step-by-step demonstration of how to perform a task or how to solve a problem is given (Sweller, 1985). Usually this leads to one or two strategies for solving particular problems. As there is a distinction, it is worth investigating the nature of problem solving. For this we turn to Jonassen who has done extensive work with problem solving. Most generic problem solving models suggest that problem solvers clarify the problem, look for and discover solutions, choose one and try it out, and then determine the effectiveness of the solution. Jonassen (2003) prefers more specific methods of problem solving that are tailored to each domain. He would argue that generic models like these are weak because they underestimate the role of domain knowledge and the ability to recognize patterns within those domains (Jonassen, 2010). For Jonassen, a more appropriate approach for math would be Polya’s recommendation of a general four-step method specifically for mathematical problems. Steps would include: understand the problem (what’s being asked, look for more information); discover solutions (look for patterns, organize info); carry out the plan; and, finally, evaluate its effectiveness (Jonassen, 2010). For Jonassen, problem solving, as a process, requires two attributes. The first is that problem solving requires the mental representation of the problem known as the ‘problem space,’ ‘problem schema,’ or ‘mental model’ of the problem (Jonassen, 2010). For this paper we will refer to mental models and problem spaces. As one “develops solutions,” one of the most critical problem-solving processes is the construction of a mental model of the attempted problem. He emphasizes the importance of constructing mental models of the problem in order to understand the relevant and irrelevant elements of the problem and how they interact. A viable solution can only
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be obtained when the problem solver develops a correct model of the problem in its context (aka domain) (Jonassen, 2010). He would further argue that understanding is dependent upon these mental models that are developed throughout the “developing solutions” phase. Secondly, he contends that problem solving requires manipulation and testing of the mental model of the problem in order to develop possible solutions. He claims that understanding and far transfer is improbable if a strong mental model is not fashioned in this manner (Jonassen, 2010). Excel spreadsheets allow users to accomplish both of these objectives, turning “plug and chug” into several strategic mental models for the user to activate. Excel spreadsheets help to accentuate the developing solutions phase of Problem Solving, making this tool much more robust and defined, as it serves as a catalyst for a better identification of the problem space. Transfer Transfer is defined as the ability to transmit what has been learned in one context into another (Bransford, 2000). One can see how this would be beneficial for problem solving purposes. As math educators, this concept is a major goal of the problem solving processes being taught. Many students, over time, become highly proficient in one or two methods for solving problems from the learned context. The dynamic modeling tools mentioned earlier allow learners to represent problems by providing them with the ability to construct the problem space modeling processes. More specifically, the Excel spreadsheet became an effective mindtool that embedded its likenesses into the user. In other words, the more you use the tool, the more you develop mental modeling processes like the tool (Jonassen, 2006). Jonassen would argue that students fail to transfer problem-solving skills because they have not sufficiently developed a qualitative mental model to represent problems (Jonassen, 2003). He argues, “If a problem solver understands how to solve a problem in only one way (for example, through a procedure), then he or she will not be able to transfer problem-solving skills successfully” (Jonassen, 2003, p. 86). Described another way, experts know their domain in a variety of ways, and as the problems change, they are able to adapt to new contexts due to strong mental models. In contrast, novices have one brittle mental model that is limited to the immediate context of the problem they are solving (Vosniadou, 2008). These expanded mental models could be a result of the interaction between the modeling tool and the mental models formed (Jonassen, 2003). The Excel spreadsheets as mind-
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tools were transferred into the users. As a result they were better able to manipulate and solve the problems on the assessment. Bransford (2000) characterizes transfer as being affected by the context of original learning. In other words people can learn in one context, but then fail to transfer their knowledge to other contexts. He offers examples such as a group of Orange County homemakers who succeeded in making supermarket best-buy calculations but performed poorly on equivalent, traditional, paper-based mathematics problems (Bransford, 2000; Lave, 1988). Another example include Brazilian street children who relied on math skills when making sales in the street but were unable to answer similar problems presented in a school context (Bransford, 2000; Carraher, 1986). The implications of these examples follow the research, which indicates that when a subject is taught in a single context rather than multiple contexts, transfer becomes difficult (Bjork, Richardson-Klavhen, & Izawa, 1989; Bransford, 2000). Transfer High Road or Low Road Johnson (1995) describes two broad categories of transfer in Transfer of Learning, which draws on the work of Clark and Voogel as well as Perkins and Salomon (Clark & Voogel, 1985; Johnson, 1995; Perkins & Salomon, 1988). ‘Near transfer’ refers to students’ application of knowledge and skills in situations and contexts that are very similar to those in which the learning originally takes place (Johnson, 1995). An example of this would be if a person normally drives a manual Honda Civic but had to drive an automatic Toyota Corolla. They would be able to easily accomplish this task without a thought. The tasks of driving the two cars are “near” to each other; so the driver can adapt to the new car fairly easily (Wilhelm, 2008). ‘Far transfer,’ however, involves progressing generalizable skills in order to be used in a different context. For example, if this same person were to drive a tractor that has three clutches, their thinking processes about driving would have to rely on conceptualization. She must be able to see the connections between this new task and what she knows about driving to adapt to this new situation and effectively apply that knowledge. To drive the tractor, she would have to achieve far transfer (Wilhelm, 2008). Far transfer requires students to deliberately analyze a new or unfamiliar situation, retrieve concepts and then apply them to the new context (Salomon, 1988). Furthermore, students must be able to look beyond the surface features of the content and recognize abstract rules that apply.
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For a few authors, the terms near and far are replaced with the terms “low road and high road” (Johnson, 1995). Perkins and Salomon explain this as they speak of transfer occurring by way of two different mechanisms. “Low road transfer” involves the ability to generate practiced procedures for conditions that are very close to the context in which they were learned. “High road transfer” requires ‘mindful abstraction’ from the context of learning as well as a deliberate search for connections (Perkins & Solomon, 1992). This kind of transfer further demands time for the participant to explore the problem by utilizing deliberately invested mental effort (Perkins & Salomon, 1992). The Purpose of the Study A pilot assessment was given to determine if there was a significant difference in performance between students who had successfully completed first year Algebra and current first year algebra students. Students who received a year of Algebra instruction performed better than those who were currently taking Algebra on many assessment items. However, there was a lack of improvement evident in three learning
targets. Those learning targets were to approximate and interpret rates of change from graphical and numerical data, analyze functions (by investigating rates of change, intercepts, zeros, and local/global behavior) and write equivalent forms of equations and systems of equations and solve them with fluency. In the case of three assessment items, students who completed Algebra instruction performed worse than those who have not. The three learning targets mentioned represent the majority of fundamental concepts explored in a first year algebra course. The assessment data support the conclusion that although the curriculum demonstrated adequate breadth and depth of study, students fail to exhibit mastery of fundamental algebraic concepts. In response, the purpose of this study is to compare the effectiveness of two instructional practices at a High School, on first year algebra student’s ability to model real-world phenomenon algebraically. Research Question What is the effectiveness of the use of Excel spreadsheets, in an Algebra unit, compared to traditional methods?
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Methodology Students (N=53) from four of fourteen sections of first year algebra classes at a Midwestern High School were eligible to participate in this study. Each course section was composed of freshman that had no prior algebra instruction. Two teachers were involved in the study, each teaching two of the four sections. Four homogenous sections from two teachers were selected based on data from a local common assessment. These sections exhibited similar competency in first semester concepts. An analysis of variance was used to ensure homogeneity of the four identified sections. The test yielded a p = 0.77 suggesting that for our study there may have been no difference in mean preintervention assessment scores for the students who had different teachers and classes. There were no unusual criteria that might have lead to non-equivalent classes. Using classes with similar characteristics allowed for comparison of the two teaching methods within normal instruction. Each teacher used the experimental method for one class as their second class received instruction consistent with the traditional instructional practices used to teach this content. The majority of problems, activities, and assignments were the same for both treatment and control groups in this study. The intervention was in the treatment groups’ use of Excel to explore problems and the control groups’ use of only traditional methods such as graphing calculators, textbook activities, and paper and pencil methods. Both teachers kept logs and met regularly to ensure consistency and to minimize the possibility of confounding the results. Although this is a two-by-two factorial design, the teacher effect should be mitigated due to each educator teaching one experimental and one control group. Participants Consent was obtained to collect data from 53 first year algebra I students in four classes. The students ranged in age from 14 to 16 years old. The principal investigator was unable to obtain consent from five students distributed across all four sections. To address the issue of volunteerism, a second ANOVA confirmed the homogeneity of the four sections less the five students who did not consent. The test yielded a p = 0.76 indicating that it was still that there was no difference in the assessment scores across sections. None of the students had enrolled in an algebra course prior to this school year. The ability levels of the students within each section varied sig-
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nificantly as indicated by local assessment scores. None of the participants were classified as special education or had individualized education plans. Similarly, none of the students were classified as gifted by district tests. The demographics of the students were diverse. The genders of the students consisted of 24 female students and 30 male students. The ethnic makeup was 66.5% African American, 22.3% Hispanic, and 11.2% Caucasian. Materials All four classes used activities from Carnegie Learning’s Text Algebra I. Both classroom teachers met daily during the period of the study to ensure the instruction received by the students was consistent in each class. During this investigation the control group completed three sections from Carnegie Learning’s Text Algebra I. In addition to truncated versions of the activities from the textbook, the treatment group explored three activities in which they were asked to use Microsoft Excel to create a tabular, symbolic and graphical representation of a scenario. These activities were adapted from the “Supreme Court Handshake” and “Beyond Handshakes” lessons on the NCTM’s Illuminations website (National Council of Teachers of Mathematics). Measurement Instrument The principal investigator developed an 8 item criterion assessment to assess the students’ ability to identify the appropriateness of linear vs. quadratic functions, represent relationships symbolically, evaluate functions, and choose appropriate graphical representations of a mathematical relationships. This assessment comprised of both multiple choice and open-ended responses. A cover sheet was included with the assessment directing students to show their work they used to arrive at an answer and graphing calculators were used. The dichotomously scored assessment was scored using an optical device such as ParScore or Scantron. For class credit purposes, partial credit was usually awarded to students who showed a correct strategy with incorrect arithmetic, but for this study, these types of responses were marked as incorrect. The assessment was previously validated by administering it to a group of 28 high school sophomores enrolled in a college prep geometry class. Guidelines for selecting assessment items suggest that each question
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have a difficulty index greater than 0.16 but less than 0.84. In addition, the discrimination index for each item should be greater than +0.30 (Nitko & Brookhart, 2007, p. 331). All of the discrimination indexes were greater then +0.30. The KR20 reliability coefficient reported in the computer scoring output was 0.71, suggesting that the assessment had an acceptable reliability. This is a low but acceptable alpha which makes it more difficult to find a true difference if it exists. Procedures Once homogeneous sections were identified from a larger pool of fourteen classes, their respective classroom teacher read students a recruiting script. It was explained that the principal investigator was interested in comparing two different instructional strategies in an attempt to improve the teaching and learning of Algebra I. A letter to the students and parents along with two copies of an informed consent form was given to the students to take home. They were instructed to return one signed copy to their classroom teacher and retain one copy for their records. All students, regardless of consent, took part in either the control or experiential activities since the instructional variations were considered to be within the parameters of the normal instructional program. The data from students who returned signed informed consent documents are the only data reported in this analysis. The first treatment class section was determined at random using a pseudo random number generator. The remaining section, taught by the teacher of the selected class, was assigned to the control strategy. Similarly, a section taught by the second teacher was selected at random to receive the treatment and the remaining section received the control strategy creating a complete 2x2 design. This design ensures that every possible combination of teacher and instruction delivery method was present in the experiment. When students used the spreadsheets to model algebraic relationships, they created a table of values using Excel’s formula feature. Students explored the relationship using smaller values and then deduced the recursive relationship, a form of elaboration. For example, in the “Supreme Court Handshake” problem (“National Council”, n.d.), students needed to use the recursive relationship
to find the number of handshakes for a
greater numbers of people. Figure 1 shows one of the ways in which students used recursive relationship to write formulas. Excel spreadsheets calculated the output of the formula reducing the amount of tedious calculations required of the student. Additionally, students were able to easily populate lists of inputs using formulas as well Figure 1.
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Figure 1. Handshake Representation in Excel Sheet. When students were confident that their formulas were correct, they used the “fill down” feature in Excel to insert formulas in subsequent cells. Excel automatically changes the cell references to those that represented the appropriate inputs or number of people Figure 3. Lastly, students created graphical representations to observe the long-term behavior of the function. Students identified the domain and range values of the function and the appropriate quantity labels (Also Figure 2). Table 1 shows a day-by-day comparison of the activities performed by the treatment and control group.
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Figure 2. Excel Formulas and Graph for Handshakes. Table 1 Treatment Comparison Time
Control
Treatment
Processing Differences
Day 1
8.1 Website Design: Introduction to Quadratic Functions (Problem 1)
8.1 Website Design: Introduction to Quadratic Functions (Problem 1)
None
Day 2
8.1 Website Design: Introduction to Quadratic Functions (Problem 2)
Supreme Court Handshake (Excel)
Students create their own inputs and students generalize the relationship symbolically to utilize the formula feature of Excel.
Day 3
8.2 Satellite Dish: Parabolas
Beyond Handshakes: Triangular Numbers (Excel)
Students are required to use recursive relationships to complete table of values in Excel
Day 4
8.3 Dog Run: Comparing Linear and Quadratic Functions (Problem 1)
8.3 Dog Run: Comparing Linear and Quadratic Functions (Problem 1)
None
Day 5
8.3 Dog Run: Comparing Linear and Quadratic Functions (Problem 2)
Beyond Handshakes: Pyramidal numbers (Excel)
Students are required to use recursive relationships to complete table of values in Excel.
Day 6
Assessment
Assessment
High-road transfer takes place for students who used Excel
Before students modeled phenomena using Excel, they showed some mastery of symbolic manipulation using the properties of equality and operations with rational numbers. Once students demonstrated fluency solving equations and operations with rational numbers, the students began organiz-
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ing and representing relationships in a tabular form on a spreadsheet. Excel’s formula function was used to recursively compute the value of dependent quantities. With some guidance, students began to generalize recursive relationships using explicitly defined functions. Subsequently, students used their knowledge to express relationships graphically. At this point, the students were able to represent linear relationships using algebraic equations. They then begin to calculate output values of functions using the spreadsheet to handle the routine computation. The last concept the students explored was simultaneous graphs of linear functions. They explored the effect of different coefficients on the behavior of the graph. Last, students used what they knew about modeling linear relationships and began to explore other types of relationships using Excel to model the relationship. The assessment consisted of a 50-minute quiz utilizing multiple choice and open-ended responses. Students were prompted to show work where appropriate. Results The means and standard deviations for control and treatment groups of both teachers are reported in the Table 2 below. Table 2 Teacher Environments, Means and Standard Deviations Environment
Teacher
Mean
SD
Traditional
Teacher 2
3.08
2.11
Traditional
Teacher 1
3.25
1.42
Excel
Teacher 2
4.65
1.73
Excel
Teacher 1
4.75
1.66
A two-by-two ANOVA of posttest data was used with the factor Teacher (Teacher 1, Teacher2) and Treatment (Excel & Traditional). The main factor, Teacher, was not significant, F(1,49) =.077, p = .782. For the factor of Treatment we see F(1,49) = 9.979, p = .003. This indicates that the mean of the Excel Treatment group, M= 4.7, was significantly greater than
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the mean of the Traditional Control group where M = 3.2. The interaction of teacher and treatment was not significant as F(1,49) = .004, p = .948 (See Figure 5).
Figure 5. Interaction of teacher and treatment. Table 3 Test of Between-Subject Effects Source
Type III Sum of Squares
Df
Mean Squares
F
Sig.
Corrected Model
30.70*
3
10.23
3.36
.03
Intercept
801.25
1
801.25
262.97
.00
Teacher
.235
1
.24
.077
.78
Treatment
30.39
1
30.39
9.98
.00
Teacher*Treatment
.013
1
.01
.00
.95
Error
149.30
49
3.04
Total
1028.00
53
Corrected Total
180.00
52
Note: R2 = .171 (Adjusted R2 = .120)
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Although our study consisted of a small group of students our data shows that teacher and treatment was not significant, which leads us to infer that the Excel spreadsheet method was responsible for the students in the treatment groups significant out performance of the traditional control group students. Discussion This experimental investigation considered the effects of dynamic modeling in an algebra classroom. Students who participated in the Excel treatment group performed significantly better on the assessment than did students in the control group. The assessment presented questions that involved different contexts than those in which the students learned. In essence, students who performed better were able to transfer their understanding in a deep and meaningful way. It is, then, reasonable that, for our study, far transfer was able to occur because of strong mental models (Jonassen, 2010). These results suggest that the Excel instrument, as an instructional tool for mathematics, helped to form significant mental constructs for deeper understanding of content. During the “discovering solutions” phase of problem solving, the Excel spreadsheets provided students with opportunities to examine symbolic representations and required them to go beyond the information given to develop skills that involve justification or explanation. By enhancing the problem space, the spreadsheets allowed students to acquire several strategies. This gave them an overwhelming advantage over mere plug and chug techniques used by the control group. Both educators interpretively reflect that students in the treatment group created tables that were analogous to the spreadsheets created in class. These reflections report that the experimental group of students used their tables in a variety of ways to identify the quantities that were held constant, variable quantities, and recursive relationships. This suggests that, within this study, students used the Excel spreadsheet as a mindtool and developed a significantly different understanding of the mathematics content compared to the traditional groups. This finding aligns with Arnoux and Finkel (2010), as they suggest that understanding math content depends on building internal (mental) representation that is sufficiently adapted to the object. Furthermore, they believe that the more a student is able to build mental representations, the more he is able to think effectively. (Arnoux & Finkel, 2010). This level of understanding then makes it easier to reach effective levels of transfer.
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According to Perkins and Salomon, transfer of learning requires teaching explicitly for transfer, that is, teaching students to go beyond regurgitation to practice mindful abstraction and reflective thinking (Perkins & Salomon, 1992). This is done in order to develop deeper understanding and help students make connections they otherwise cannot with their existing mental model. Therefore, instruction must include helping students develop habits that foster strong mental models (Perkins, 1993). In contrast, a lack of strong mental representations leads to short-term gains in mathematical understanding and hinders the process of transfer. Comparing Excel and traditional instructional methods, although they are equivalent, may seem intuitively problematic. One may claim that the Excel method was far superior to the traditional method implying that this study heavily favored the Excel spreadsheet use. However, both teachers had a high stake in choosing instructional methods. As Subject Matter Experts they identified both methods, traditional and treatment, as instructionally sufficient for the given assessment. As public school educators, the two participants had a professional obligation to provide quality opportunities for their students to succeed. The educators placed student performance above all other considerations, as their students’ performances constituted permanent records of content knowledge. Therefore, the educators agreed that both methods were equally adequate. This study further shows that statistically the main factor, Teacher, was not significant, F(1,49) =.077, p = .782. This, in turn, shows that the study compared two equal methods in order to determine their relative effectiveness towards aspects of functions in Algebra I. Implications for Instruction Spreadsheets are an ideal instructional tool for learning mathematics, as they are excellent simulators of mathematical content and are relatively easily accessible. According to Jonassen (2010) simulations, like Excel, are environments where learners manipulate components in such a way that the main task of the learner is to infer, through experimentation, characteristics of the model underlying the simulation (Jonassen 2010; De Jong and Van Joolingen, 1998, p. 179). “When learners interact with the simulation, they change the values of (input) variables and observe the results on the values of other (output) variables. These exploratory environments afford learners the opportunity to test the causal effects of factors” (Jonassen, 2010).
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Therefore, if the purpose of instruction is to develop the transfer of knowledge through developing problem- solving skills, then Excel spreadsheets are more beneficial than traditional methods. We believe the Excel spreadsheet performed as an effective mindtool, allowing users, in this study, to obtain more robust mental strategies for solving the problems. For Excel spreadsheets to be effective, the instructor must view teaching as a process of developing and enhancing students’ ability to go beyond overly dependent on context. Instead the instructor’s must view their role as a facilitator for learning. Educators must help learners choose, adapt, and invent tools for solving problems in order to facilitate transfer (Bransford, 2000). Through this lens the educator will seek technologies that will limit tedious calculating loads and allow them allow them to use cognitive tools to explore and infer. As a result students will be able to develop skills that require stronger mental models for justification or explanation. As this was a small group of students this provided a reasonable detraction from generalization as the findings can only be referenced to this study. However, future research studies could investigate the depth to which the Excel spreadsheets allow the content to transfer. It could answer questions such as “Are students able to utilized skills learned in a real life situation”, granted that this study utilized a rather limited assessment of students abilities. A further research investigation could explore a longer test to gather more information on the specific aspects of function analysis that were mastered. As this experiment did not inquire about the metacognitive processes involved in the transfer process, another study could investigate the role metacognition played in the transfer processes of mindtools. References Arnoux, P., & Finkel, A. (2010). Using mental imagery processes for teaching and research in mathematics and computer science. International Journal of Mathematical Education in Science and Technology, 41(2), 229–242. doi:10.1080/00207390903372429 Bjork, R. A., & Richardson-Klavehn, A. (1989). On the puzzling relationship between environmental context and human memory. In C. Izawa (Ed.), Current issues in cognitive processes the Tulane Flowerree symposium on cognition (pp. 313–344). Lawrence Erlbaum Associates. Retrieved from http:// psycnet.apa.org/psycinfo/1989-98139-008 Bransford, J. (2000). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press. Carraher, T. N. (1986). From drawings to buildings; working with mathematical scales. International Journal of Behavioral Development, 9(4), 527-544.
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Clariana, R. B., & Strobel, J. (2007). Modeling technologies. In M. J. Spector, M. D. Merrill, J. Merrienboer, & M. P. Driscoll (Eds.), Handbook of Research on Educational Communications and Technology (329-342). New York, NY: Taylor & Francis Group, LLC. Clark, R. E., & Voogel, A. (1985). Transfer of training principles for instructional design. ECTJ, 33(2), 113-123. De Jong, T., & Van Joolingen, W. R. (1998). Scientific discovery learning with computer simulations of conceptual domains. (C. A. Grant, Ed.) Review of Educational Research, 68(2), 179–201. doi:10.3102/00346543068002179 González-Calero, J. A., Arnau, D., & Puig, L. (2013). Solving word problems algebraically in a spreadsheet environment in a primary school. Research in Mathematics Education, 15(3), 305-306. Hiebert, J. & Carpenter, T. P. (1992). Learning and teaching with understanding. In Grouws, D. A. (Ed.). Handbook of Research on Mathematics Teaching and Learning (pp. 65-97). New York, NY: Macmillan. Izawa, C. (1989). Current issues in cognitive processes: the Tulane Flowerree Symposium on Cognition. Hillsdale, NJ, England: Lawrence Erlbaum Associates. Jonassen, D. H. (2003). Learning to solve problems: An instructional design guide. San Francisco, CA: Pfeiffer. Jonassen, D. H. (2006). Modeling with technology: mindtools for conceptual change (3rd ed.). Upper Saddle River, NJ: Pearson. Jonassen, D. H. (2010). Learning to solve problems: A handbook for designing problem-solving learning environments. New York, NY: Routledge. Johnson, S. D. (1995). Transfer of Learning. Research Brief. Technology Teacher, 54(7), 33-35. Lave, J. (1988). Cognition in practice: Mind, mathematics and culture in everyday life. New York, NY: Cambridge University Press. Lavidas, K., Komis, V., & Gialamas, V. (2013). Spreadsheets as cognitive tools: A study of the impact of spreadsheets on problem solving of math story problems. Education and Information Technologies, 18(1), 113-129. National Council of Teachers of Mathematics (2008). Algebra: what, when, and for whom: A position of the National Council of Teachers of Mathematics. Retrieved from: http://www.nctm.org/about/content.aspx?id=16229 National Council of Teachers of Mathematics (n.d.). Illuminations: Supreme Court Handshake. Retrieved from Illuminations: http://illuminations.nctm.org/Lesson.aspx?id=2112 Nitko, A. J., & Brookhart, S. M. (2007). Educational Assessment of Students (5th ed ed.). Upper Saddle River, NJ: Pearson. Ntsohi, M. M. (2013). Investigating teaching and learning of Grade 9 Algebra through excel spreadsheets: a mixed-methods case study for Lesotho (Doctoral dissertation, Stellenbosch: Stellenbosch University). Perkins, D. N., & Salomon, G. (1992). Transfer of learning. International encyclopedia of education, 2. Retrieved from http://learnweb.harvard.edu/alps/thinking/docs/traencyn.htm
Excel Spreadsheets for Algebra: Improving Mental Modeling
427
Qin, Y., & Simon, H. A. (1992). Imagery and mental models in problem solving. In Proceedings of the AAAI 1992 Spring Symposium on Reasoning with Diagrammatic Representations, Palo Alto, CA. Salomon, G., & Perkins, D. (1988). Teaching for transfer. Educational Leadership, 46(1), 22-32. Salomon, G., & Perkins, D. N. (1989). Rocky roads to transfer: Rethinking mechanism of a neglected phenomenon. Educational Psychologist, 24(2), 113-142. Sweller, J. (2006). The worked example effect and human cognition. Learning and Instruction, 16(2), 165–169. Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2(1), 59-89. Tabach, M., Hershkowitz, R., & Dreyfus, T. (2013). Learning beginning algebra in a computer-intensive environment. ZDM, 45(3), 377-391. Vosniadou, S. (2008). International handbook of research on conceptual change. (Stella Vosniadou, Ed.) Learning Vol. 1, p. 740. Wilhelm, J. D. (2008). The problem of teaching for transfer: taking the low road or the high road? Next Steps in the Journey, 15(4), 45–47. Wu, Y., & Wong, K. Y. (2007). Impact of a spreadsheet exploration on secondary school students’ understanding of statistical graphs. Journal of Computers in Mathematics and Science Teaching, 26(4), 355-385.
