Communication through Mathematics: The Effects on Mathematic Reasonableness

Abstract: 

This Action Research focused on the efficiency of using prescribed tools to aid in justification and reasonableness in solving mathematical problems. A lack of connection to real life and mathematical situations in elementary schools was noted in the observations. This study was initiated based on the hypothesis that verbal and pictorial justifications will improve mathematical reasonableness. Pre-instructional samples, assessments, and observations showed little to no reasonableness in students’ formulated answers. Students were having difficulty connecting math in real life and forming accurate answers. Post-instructional samples, assessments, and observations showed a significant increase in justification as well as statistically significant differences in posttest scores on the mathematics concept the students were learning during the study. The use of a progressive approach to learning, verbal, and pictorial justifications accurately transferred to the students’ computation skills.

Table of Contents: 

    Introduction

    This research stemmed from my interest in studying the relationship between verbal and pictorial justifications and reasonable explanations of mathematics. As a pre-service teacher in a third grade classroom that heavily stresses efficient testing strategies, it was obvious from cursory observations that students used few, if any, efficient strategies, such as drawing and conversing about the methods they were using in order to solve mathematical problems. Observations of students in this setting showed that students had difficulty relating test questions to real life experiences. Students could procedurally solve problems but showed limited reasoning in reaching the conclusions. The limited use of drawing and explaining were absent from the students’ strategy set, creating a problem of reasonable mathematics justification that prompted the investigation.

    Purpose

    The purpose of this Action Research was to examine third grade students’ understanding of the mathematics concept of fractions as related to their reasonable representation of the concept with pictures and verbal justifications.

    Question

    What is the effect of mathematical tools support (verbal and pictorial justifications) on students’ understanding of the concept of fractions?

    Hypothesis

    It was hypothesized that the use of verbal and pictorial justifications would positively relate to third grade students’ understanding of the mathematics concept of fractions.

    Literature Review

    How do we communicate mathematics thought? One way to enhance a student’s ability to express mathematical understanding is to provide mathematical terminology and accurate pictorial representations that represent mathematical thoughts (Friedman, Kazerouni, Lax, & Weisdorf, 2011). For effective communication in mathematics, students can be taught a solid foundation of mathematical terminology to understand concepts. Prompting students with questions is a process that can aid in their cognitive learning. Open-ended questions elicit both discussion and investigation in order to build a foundation for students (Friedman et al., 2011). Denise Forrest (2008) defines verbal communication as “a medium for expressing thoughts and feelings” (p. 24). This type of social interaction allows students to create interpretations of the world around them. Building a framework for verbal interaction plays a significant role in the success of the learning-teaching process. Diezmann and McCosker (2011) define pictorial justifications as self-created representations that give teachers insight into mathematical thinking.

    When students draw an image of the cognition applied to solving a mathematics problem, teachers gain insight into how to foster mastery of the mathematics concept. According to Moyer-Packenham, Ulmer, and Anderson (2012), student-generated justifications through pictures provide students with opportunities to demonstrate their thinking. The authors studied the effects on pictorial representations and student learning wherein students were engaged in a fraction unit. Moyer-Packenham et al.’s (2012) research particularly focused on pictorial representations that students used for fraction learning, and showed that students had “opportunities to represent their mathematical knowledge in a variety of ways through physical manipulatives, pictorial representations, and symbolic notation (Moyer-Packenham et al., 2012). In conclusion, the authors found that students were able to develop a deeper understanding and visualization of fractions through the use of pictorial models. Students were also able to achieve better results on test questions pertaining to pictorial representations. Similar procedures were taken in this Action Research using the third grade students.

    Diezmann and McCosker (2011) conducted a study of student mathematical understanding that used numerous strategies, including verbal and pictorial justifications, as well as an abundance of group and partner work. The type of collaboration used in this study allowed students to check and understand one another’s work. The authors found that students expand their knowledge when they are provided time to share and discuss their representations. These indicators of thinking showed that students were able to give adequate explanation of their representations, with each child producing a representation of his or her thinking, when allowed to work in pairs to complete tasks.

    Salter, Pang, and Sharma (2009) show how teaching time should be shifted from traditional lecture, which is designed to cover course content, to a more active learning approach.  Salter et al. (2009) demonstrates that using learning tasks and feedback online allows class time to be shifted toward more conversation, coaching rather than instruction, and feedback among peers and teachers. A more conversational approach to teaching can guide students to be confident enough to use verbal and pictorial justifications without the fear of failing or producing the wrong answer.

    Action Research Format

    Action Research is a form of research that seeks to assist teachers in knowing more about teaching and learning in individual classrooms based on research that is conducted in the classroom by the teacher/pre-service intern. Findings from Action Research studies assist teachers as they work to improve the way they address issues and solve problems concerning teaching and learning in a single classroom. This process is a way to understand teaching practices and make plans accordingly to improve practice. In this research, the mentor teacher and the pre-service teacher collaborated to create the study and conduct the research, which required coordination between researchers through shared learning, documentation, and reflection. 

    Method

    Setting and Population

    This study was conducted in a Title I bilingual campus consisting of 11.6% African American students, 31.5% Hispanic students, 54.5% White students, 0.4% Native American students, and 2.0% Asian students. The campus was defined by various economic and linguistic factors, such as 50.5% economically disadvantaged students, 17.9% Limited English Proficient students (LEP), 34.6% At-Risk students, and 10.9% mobility students.

    In the third grade classroom where the Action Research was conducted, 34.7% of the students were African American, 8.7% were Hispanic, 47.8% were White, 4.3% were Native American, and 4.3% were Asian. The classroom setting was interactive and engaging during the course of this Action Research. Meaningful conversations were constantly flowing throughout the room. Students were allowed to work in pairs, which helped them feel comfortable in the classroom. Some worked on the carpet with clipboards; others were spread out on the floor with construction paper to draw pictures of mathematics concepts. Students who did not work well with the noise in the classroom were allowed to work in the hallway with a partner. Students’ drawings were illustrated on construction paper through markers or notebook paper through pencils. The drawings were representations of the problems at hand, consisting of shapes cut into fractional parts, houses with windows to describe fractions, and dots put into sets to create fractional groups. Conversations about the drawings took place between partners as well as teachers who observed, listened, supported, and questioned students’ reasoned explanations.

    Instrumentation

    Study Island. Study Island was developed by two entrepreneurs, computer scientist Cameron Chalmers and marketer-economist David Muzzo, who were passionate about developing an online education program that would help learners of all ability levels. First launched in 2000, Study Island was widely accepted based on results that showed students’ improvement in core academic areas. The software provides standards-based assessment, instruction, and test preparation. Study Island is used to advance student achievement through digital learning. The programs and lessons are integrated with traditional classroom instruction to provide students with additional time and sense of control over their learning experiences. Each Study Island session focuses on an academic standard or underlying topic, setting goals to master one concept at a time. Immediate feedback and explanations for each question help students take responsibility for their own learning. Although Study Island is supplemental and not used as the primary curriculum, there are mini-lessons for review either before or during each individual standards-specific set of questions. This allows students to refresh quickly before assessing knowledge.

    Several studies of the software show that Study Island is an effective way to measure student success in mathematics. A case study research of Study Island in Texas shows students’ yearlong progression, particularly in the area of mathematics (Magnolia Consulting, 2008). The research also showed that schools using Study Island had higher testing results than schools not using Study Island. Styers (2012) noted effect sizes ranging from 0.29 to 0.95 when standards alignment, modifying instruction, providing feedback, and individualized support were studied in a controlled environment.

    Reasonableness Observation Protocol. This tool was a self-made instrument used to observe students as they illustrated and conversed about fractions while working on problems during the students’ ten-minute work time. The instrument consisted of three categories: pictorial justifications, verbal justifications, and reasonableness. All three categories were judged in a dichotic manner: present/not present for the first two categories and reasonable/not reasonable for the third category. The protocol was used once each week as a means of tracking students’ ability to represent their thinking about solving fraction problems, hence demonstrating their understanding of the concept.

    Procedure

    Assessment. This Action Research was conducted over a six-week period. The first and sixth week were used for pre- and post-assessments on Study Island. These assessments were based on Computer Adapted Testing (CAT) standards for each student. The pre- and post-assessments consisted of different material. While the pre-assessment focused on basic fraction concepts, the post-assessment was more rigorous and in-depth, building up from pre-assessment material. The students were expected to show knowledge of fractions represented on number lines, understanding part of whole, equivalent fractions, and comparing two fractions based on pictorial models. Using Study Island, the students completed ongoing assessments once a week on fraction concepts. With the exception of one week in the six-week research period when school was not in session due to inclement weather, students completed ongoing assessments on Study Island. These ongoing assessments included topics of fractional parts, equivalent fractions, and comparing fractions. These were completed individually with immediate feedback. Students worked individually on Study Island lessons, practice questions, and assessments based on fraction concepts. Upon completion of the six-week Action Research, Study Island was used to give a post-assessment.

    Intervention. The fraction lessons lasted 30 minutes each, consisting of 10-15 minutes of introduction by the teacher with the whole group, 10 minutes working in groups or partners, and 5-10 minutes of closing and reflection led by the teacher with the whole group. During the student work time, the pre-service teacher observed students working through problems. Students were engaging in meaningful conversations and using many different pictorial representations. Once a week, the Reasonableness Observation Protocol was used during observations of students as they were drawing, discussing, and reasonably working through assigned problems.

    Data Collection

    Data were collected employing two instruments. Data were collected in the first week of the study to establish baseline. Students worked on Study Island to complete a pre-assessment, ongoing assessments of fractional parts, equivalent fractions, and comparing fractions, and a post-assessment of fractions. Data that showed each student’s performance level, noted in the form of percentages, were collected from Study Island each week.

    The pre-service teacher collected data of group reflections and conversations among students. These data were collected to indicate when students were using drawings, verbalizing thoughts, and creating reasonable justifications. Observations were documented on a three-point system shown in the reasonableness chart. Once a week, the pre-service teacher observed the students to see if they were reasonably verbalizing pictorial justifications of fraction concepts.  Notes were taken regarding students’ ability to draw pictures and verbalize, but not necessarily show reason in the justification.

    Data Analysis

    This research began with 23 students in the study. After adjusting for absences and other factors contributing to aberrances, data from 14 students were used for analysis.

    Pre/Posttest Analysis

    The analysis, presented in Table 1, shows that students performed statistically significantly different, at the p < .05 level on the posttest (Table 3) following four weeks of drawing, conversing, and providing reasoned justifications for fraction understanding.

    Observation Protocol Analysis

    A trend analysis of the observation protocol data, presented in Table 2 and Table 4, indicated that all students were showing accepted levels of performance by the fifth week. More than half of the class performed below optimal levels during Week 2 of the observation. With further exploration and instruction, Week 3 had a slight increase, resulting in 64% of students performing at optimal levels through pictorial models and 100% of the class effectively using verbal justifications as well as reasonable mathematic justifications. Week 4 showed a slight decrease in the trend, resulting in 57% of the students using pictorial justifications, 71% using verbal justifications, and 64% constructing reasonable thoughts. After observation and refocusing on the importance of using pictorial and verbal justifications to reasonably solve problems, students increased in all areas, resulting in all of the students operating at 100% in all three categories in the final week. Despite the slight decrease in Week 4, students were moving toward understanding and were able to reasonably solve mathematic problems by the fifth week.

    Figure 1 shows the marked progress of students to pictorially represent, verbalize, and reasonably represent their thoughts about fractions across the four weeks of intervention.  

    Results

    The testing analysis and perceptual data revealed that students varied in performance. Based on pre- and post-assessment data, 12 out of the 14 students increased percentage scores by at least 20%. The other two students both increased by 1-10%. All students showed marked progress in the understanding of fractions as measured by Study Island results. The variance of performance showed that students passed with a score of 80% or higher on the post-assessment. The implication is that the experiences of drawing, talking, and creating reasonable explanations for answers to questions about fractions affected student performance on tests of fraction understanding.

    Observations of students resulted in continuous progression toward reasonable responses. One example of reasonability was noted in the study of fair shares. Specifically, students were able to find fair shares when a rectangle was divided in half and to logically discuss it as a brownie cut into equal pieces to share. By being able to relate to the rectangle not only as a shape, but also as something meaningful, students were able to reasonably discuss how to make fair shares of brownies, not just paper fractions. Students discussed how to label an equal share as a fraction of a whole by explaining that one half of a brownie is one half of the whole brownie.

    When students were introduced to fractions of 12, they thought about a set of 12 items and divided fractional parts from the items they verbalized or drew. Students were observed drawing 12 dots to represent marbles, and then separating them into groups using division strategies to make fractional sets. One student stated that to create thirds, she would have to share her 12 marbles with three friends resulting in each of them having a third of the marble collection.

    Students did the same method when working with money. Students created thirds of $15.  One student drew three five-dollar bills to show a third of $15 is $5. One student also explained that if realistically she had $15 she could either have a ten dollar bill and a five dollar bill, three five dollar bills, or a combination of one dollar bills. She then stated that creating thirds of her money would be simple if she had three five-dollar bills.

    In the fourth week, students were observed struggling when working to create fractional pieces of a hexagon. They were provided hexagon templates and were able to draw, verbalize, and reasonably work to create halves of the hexagon. One student observed and stated that half of a hexagon was a trapezoid, but students struggled to create smaller fractional pieces. There was little reasonableness associated with verbalization about this concept. Students were able to verbalize their thoughts and actions, but there was no reasonableness associated with their discussion. These points of disconnect were reflected in the weekly observations wherein weekly percentages shifted to lower amounts. One notable difference was the lack of real world context for fractionalizing a hexagon in comparison to sharing a brownie or money. This may have impeded students’ success.

    Students also struggled to find many ways to make a share. For example, many students did not grasp that 1/6 + 1/4 = 1/2. When students were able to work with peers and collaboratively solve many ways to make a share, students were observed engaging in reasonable discussion. Students noted that the denominator did not always have to be the same to create shares. Students were able to relate many different pictorial images to justify and clarify their answers. Again, in this incident, the students were not supplied with real world contexts; hence, it is possible that this impeded their reasoning.

    When students worked with tangible materials, they reasonably worked through verbal and pictorial justifications. Students were read the book The Hershey’s Milk Chocolate Fraction Book by Jerry Pallotta, after which they used a Hershey bar to create a booklet of different fractional pieces of a candy bar. Along with each picture, students verbalized the fractional pieces of the candy bar. This activity gave students means to work reasonably to understand fractions. This sixth-week activity served to return the students to a relatable context for fractions, hence raising their reasonableness scores.

    During the week-by-week observation, it was apparent that students became progressively reasonable mathematicians. Initially, students were very puzzled when they tried to verbalize their thoughts. As the weeks went by, students got comfortable talking through mathematics and reasonability increased. Students incorrectly explained answers at times, but then, after thinking about what they said, revised their thinking and gave a more reasonable explanation. The highest overall level of the presence of pictures and justifications, and reasoned answers (100%) was noted in the final week of the study. It became apparent that when students had objects, paper and tools for drawing, real world contexts, and an adult with whom to visit, reasoning about mathematics emerged. With coaching from the teacher/pre-service teacher, students were able to see why their thoughts were reasonable or not reasonable. The power of hearing their words and seeing their examples served to encourage reflective practice. As opposed to their mnemonically stating everything they wrote down or drew, students provided thorough explanations with multiple examples to support answers.

    Discussion

    The purpose of this paper was to examine students’ mathematic justifications and drawings and their relationship to mathematic reasonableness. Students’ reasonableness was shown in verbal and pictorial justification as they were able to work through fraction concepts. This poses the question: Can students become increasingly reasonable mathematicians – as well as increase their understanding of the concept of fractions – in 10 minutes of exploration time each day across four weeks? As noted in the findings in this study, the answer is supported positively. According to the t-test results presented in Table 3 showing statistically significant difference from pretest to posttest (p > 0.05), the brief, daily exploration and discussion time was a sufficient amount of active learning time for students to reasonably justify fraction concepts and show gains in understanding the concept of fractions. In the 10 minutes of collaborative learning time, students were able to explore reasonable ways to solve the problems, and then were affirmed in the discussion with their peers at the end of the lesson.

    Conclusion

    The study examined whether or not the use of verbal and pictorial justifications would increase mathematic reasonableness among third grade students. The results show that the time devoted to mathematic instruction, specifically fractions, and student-centered learning had an effect on students’ learning. The methods used in this study differ from standard procedure used in third grade classrooms, poised to prepare students for state-mandated mathematics tests. Other approaches generally involve high-pressure preparation and focus on teacher-centered drill and practice exercises. The approach described in this study entrusted the learning to the students in the form of drawings coupled with discussions, which were challenged and affirmed for reasonableness in understanding the concept of fractions.

    Testing is an ever-present method of data collection in elementary schools, and teachers in third grade are particularly pressured by the state to ensure students demonstrate adequate knowledge and skills about a set of topics and concepts prior to entering the fourth grade. Teachers feel compelled to increase the pace of instruction because of time constrains and testing. With the State of Texas Assessments of Academic Readiness (STAAR) test quickly approaching, teachers feel pressured to spend more time with teacher-centered instruction and less time setting up active learning time.

    In contrast, Salter et al. (2009) explained how it is easier for students to engage in conversation among classmates if students are introduced to a topic prior to exploration and discussion. Students, especially the shy ones, are more confident if they have background knowledge about the content. Background knowledge can start with as little as an introduction before sending students off to discover among their peers. Salter et al.’s (2009) research indicates that schools should rethink course design toward a more active learning method. Results of their research show that increased interactivity and engagement for students with certain subject materials allow for meaningful learning experiences. Because of this, instructors should create a learning environment that is appropriate to increase student interactions with content, peers, and instruction, thus increasing students’ reasonability when solving mathematics problems.

    This study verifies Salter et al.’s (2009) findings. Students were given time to interact and explore, then concluded with group discussion, both verbal and pictorial, which aided in reasonability. When students were able to hear problems, content, and topics in another voice, or see things pictorially in a different image, they were able to solidify their reasonability about the problem. Immediate online feedback from Study Island coupled with small group feedback was enough to reassure and support students to continue in their process toward becoming reasonable mathematicians.

    The significance of this Action Research is that it verifies student-centered learning as a means by which to increase learning while giving students the opportunity to express their understanding of the mathematics concept of fractions. Action Research, as a method, has been demonstrated to be a valuable method for studying student growth in mathematics understanding. The internal observations conducted on students as they drew, discussed, and checked their reasonability were possible because of this method. The results lead to the potential for further Action Research to determine if this method of instruction affects the acquisition of other difficult concepts in mathematics.

    References

    • Diezmann, C. M., & McCosker, N. T. (2011). Reading students’ representations. Teaching Children Mathematics, 18(3), 162-169. Retrieved from http://www.jstor.org/stable/10.5951/teacchilmath.18.3.0162
    • Friedman, R., Kazerouni, G., Lax, S., & Weisdorf, E. (2011). Increasing communication in geometry by using a personal math concept chart. The Canadian Journal of Action Research, 12(2), 30-39. Retrieved from http://cjar.nipissingu.ca/index.php/cjar/article/view/17/16
    • Forrest, D. B. (2008). Communication theory offers insight into mathematics teachers’ talk. The Mathematics Educator, 18(2), 23-32. Retrieved from http://files.eric.ed.gov/fulltext/EJ841571.pdf
    • Magnolia Consulting. (2008). Case Study Research of Study Island in Texas. Retrieved from http://www.studyisland.com/sites/studyisland.com/files/content/research/pdfs/Case-Study-Summary-Texas.pdf
    • Moyer-Packenham, P. S., Ulmer, L. A., & Anderson, K. L. (2012). Examining pictorial models and virtual manipulatives for third-grade fraction instruction. Journal of Interactive Online Learning, 11(3), 103-120. Retrieved from http://www.ncolr.org/jiol/issues/pdf/11.3.2.pdf
    • Salter, D., Pang, M. Y. C., & Sharma, P. (2009). Active tasks to change the use of class time within an outcomes based approach to curriculum design. Journal of University Teaching and Learning Practice, 6(1), 27-38. Retrieved from http://eric.ed.gov/?id=EJ867294
    • Styers, M. K. (2012). Developing student mathematics skills: How Study Island aligns with best practice. Retrieved from http://studyisland.com

    Appendix

    Texas Essential Knowledge and Skills – Fractions. The TEKS for fractions are:

    111.5.b.(3)  Number and operations. The student applies mathematical process standards to represent and explain fractional units. The student is expected to:

    (A)  represent fractions greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 using concrete objects and pictorial models, including strip diagrams and number lines;

    (B)  determine the corresponding fraction greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 given a specified point on a number line;

    (C)  explain that the unit fraction 1/b represents the quantity formed by one part of a whole that has been partitioned into b equal parts where b is a non-zero whole number;

    (D)  compose and decompose a fraction a/b with a numerator greater than zero and less than or equal to b as a sum of parts 1/b;

    (E)  solve problems involving partitioning an object or a set of objects among two or more recipients using pictorial representations of fractions with denominators of 2, 3, 4, 6, and 8;

    (F)  represent equivalent fractions with denominators of 2, 3, 4, 6, and 8 using a variety of objects and pictorial models, including number lines;

    (G)  explain that two fractions are equivalent if and only if they are both represented by the same point on the number line or represent the same portion of a same size whole for an area model; and

    (H) compare two fractions having the same numerator or denominator in problems by reasoning about their sizes and justifying the conclusion using symbols, words, objects, and pictorial models.

    Table 1: Study Island

      Pre Assessment Fractional Parts Equivalent Fractions Compare Fractions Post Assessment
    Student 1 91.7 92.3 100.0 90.0 92.9
    Student 2 70.0 80.0 90.0 100.0 100.0
    Student 3 80.0 86.7 71.4 70.0 100.0
    Student 4 80.0 40.0 40.0 70.0 100.0
    Student 5 80.0 90.0 78.6 90.0 100.0
    Student 6 70.0 80.0 87.5 90.0 100.0
    Student 7 76.9 90.0 78.6 90.0 100.0
    Student 8 70.0 70.0 35.0 70.0 90.0
    Student 9 43.3 100.0 100.0 70.0 90.0
    Student 10 80.0 90.0 100.0 100.0 90.0
    Student 11 70.0 90.0 100.0 80.0 95.0
    Student 12 70.0 79.4 70.0 73.3 100.0
    Student 13 36.1 70.0 25.0 70.0 80.0
    Student 14 70.0 80.0 81.0 90.0 90.0

    Study Island data represent percentage correct by students at five points of assessment that were taken throughout the study.

    Table 2: Reasonableness Observation Protocol

      Week 2 Week 3 Week 4 Week 5
    Student 1

    P: 1

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    Student 2

    P: 0

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    P: 0

    V: 0

    R: 0

    P: 1

    V: 1

    R: 1

    Student 3

    P: 1

    V: 0

    R: 0

    P: 1

    V: 1

    R: 1

    P: 0

    V: 0

    R: 0

    P: 1

    V: 1

    R: 1

    Student 4

    P: 0

    V: 0

    R: 0

    P: 0

    V: 1

    R: 1

    P: 0

    V: 0

    R: 0

    P: 1

    V: 1

    R: 1

    Student 5

    P: 0

    V: 1

    R: 0

    P: 1

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    Student 6

    P: 1

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    Student 7

    P: 0

    V: 0

    R: 0

    P: 1

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    Student 8

    P: 0

    V: 0

    R: 0

    P: 0

    V: 1

    R: 1

    P: 0

    V: 0

    R: 0

    P: 1

    V: 1

    R: 1

    Student 9

    P: 0

    V: 1

    R: 0

    P: 0

    V: 1

    R: 1

    P: 0

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    Student 10

    P: 1

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    Student 11

    P: 1

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    Student 12

    P: 1

    V: 0

    R: 0

    P: 1

    V: 1

    R: 1

    P: 0

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    Student 13

    P: 0

    V: 0

    R: 0

    P: 0

    V: 1

    R: 1

    P: 0

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    Student 14

    P: 1

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    P: 0

    V: 1

    R: 1

    P: 1

    V: 1

    R: 1

    The number 1 represents the presence of pictures and/or verbal justifications.  The 0 represents non-presence. The number 1 for reasonable represents a reasoned justification and 0 represents unreasonable.

    Table 3: Results of t-test comparing pre and posttest scores on Study Island

      t df Sig.(2-tailed)
    Study Island Pretest 7.2521 12 .0001
    Study Island Posttest      

    Table 4: Weekly results of observations

      Week 2 Week 3 Week 4 Week 5
    Picture 50 64 57 100
    Verbal 50 100 71 100
    Reasoning 43 100 64 100

    The table shows students’ percentage performance based on the observation protocol data.

    Figure 1: Graphic representation of Observation Protocol