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This study provides an examination of fractional knowledge demonstrated by 25 (7th and 9th grade) students from a suburban cluster of schools in a suburban community in Georgia. Students were given five fraction problems requiring them use representations for their answers. Analysis of individual responses indicated that students at both grade levels lack a complete understanding of fractional concepts such as part to whole, fractional parts, and distances and relationships between quantities.
Research suggests that learning is more meaningful when students are given frequent opportunities to interact with different models and rethink the concepts (Dienes, cited in Post & Reys, 1979). According to Lesh, Landau, & Hamilton (1983), mathematics concepts can and should be represented other ways as well, using real-world objects, spoken symbols, written words, and written symbols. They suggest that students who use a variety of ways to represent fractions develop more flexible notions of fractions. Petit, Laird, & Marsden (2010) state using models and regularly asking students to explain their thinking plays an important role in instruction. Asking students questions as they work through problem solving helps them build upon their understanding of fractions. Heller, Post, Behr, & Lesh (1990), found that about only one fifth of seventh graders and one fourth of eighth graders have a functional understanding of proportionality.
Models should permeate instruction allowing students opportunities to problem solve and develop understanding of fractional concepts such as part to whole, fractional parts, and distances and relationships between quantities. Students demonstrate more difficulty finding the fractional part when the number of parts in the whole is equal to the magnitude of the denominator rather than a multiple or factor of the magnitude of the whole (Bezuk & Bieck, 1993). Three types of models students use to interact with, solve problems, and generalize concepts related to fractions are area models, set models, and number lines. Student-drawn area models can be effective for making comparisons of parts of wholes or locating fractions on a number line. Circle models can be used effectively to compare fractions if students consider the size of the whole and are accurate in their partitions into equal-sized parts. Combining models with manipulatives can help students focus on important features of the models and make comparisons (Petit, Laird, & Marsden, 2010).
Georgia Performance Standards
In the first grade students are expected to divide up to 100 objects into equal parts using words, pictures, or diagrams (G1M1N4). Specifically halves and fourths as equal parts of a whole using pictures and models (G1M1N4C). In the second grade students are expected to understand and compare fractions (G2M2N4). Students will model, identify, label, and compare fractions (thirds, sixths, eighths, tenths) as a representation of equal parts of a whole or of a set (G2M2N4). In the fifth grade are expected to compare fractions and justify the comparison (G5M5N4F).
According to the Georgia Department of Education, students begin to develop an understanding of fractions in the third grade. Students are able to view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students are able to use fractions to represent numbers equal to, less than, and greater than one and solve problems that involve comparing fractions by using visual fraction models and strategies.
Students develop understanding of fraction equivalence and operations with fractions in the fourth grade. They extend previous understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of fractions and the meaning of multiplication to multiply a fraction by a whole number.
Students begin extending understanding of fraction equivalence and ordering by using visual fraction models in the fifth grade. In addition, students are expected to compare fractions with different numerators and denominators, understand addition and subtraction of fractions as joining and separating parts referring to the same whole, apply and extend previous understandings of multiplication to multiply a fraction by a whole number, and understand decimal notation for fractions, and compare decimal fractions.
In this study, the following questions were posed:
How well do seventh and ninth grade students perform on fraction problems that require them to use representations?
Are there any significant differences by grade level?
Do students at the seventh and ninth grade level demonstrate an understanding of fractional knowledge?
Students from a suburban cluster of schools in a suburban community south of metro Atlanta participated in this study. In this cluster of schools, the students from three elementary schools, feed into two middle schools, both middle schools feed into one high school. The population of the high school, similar to the population of the feeder schools has a student population that is approximately 1% Asian-Pacific Islander, 3% Hispanic, 50% African American, and 46% Caucasian. For the 2009-2010 school year, both elementary and middle schools met adequate yearly progress (AYP). The high school did not meet AYP.
This study included students in the 7th and 9th grade. Teachers were solicited who had a professional relationship with the researchers. The classes represented included regular mathematics for students in the 7th grade and Mathematics 1 for students in the 9th grade. See Table 1 for student demographics. Of the two teachers that submitted responses, fourteen 7th grade and eleven 9th grade assessments were submitted. For the purposes of this study, responses from all 25 will be included in the commentary. Images of unique and questionable responses will be provided to support and document the student work that depicts effective instruction and learning or whether educators and other stakeholders need to focus/alter instruction to promote student learning & understanding.
A computational fraction test was developed and adapted from an EasyCBM, 6th grade probe and was administered to all participants. The test assesses fractional knowledge skills and consisted of 5 items requiring students to use representations for their answers. The assessment was designed to identify their ability to use representations, models, or embodiments of rational number quantities to answer a variety of questions (Petit, Laird, Marsden, 2010). The assessment focused on four key strategies or models students must be competent in using when working with fractions or rational number quantities: models/picture based images, symbolic representations of fractions (in fractional form), oral & written communication or fractional quantities, and using real-world objects in context (Petit, Laird, Marsden, 2010). The first question asks a question for students to determine the greater of two fractional quantities - usage of models or any representation is not expected. The number line represents the quantity or spectrum of values between 0 and 1. The number line is divided into four equal segments without the values of each segment labeled. The second question seeks to determine if students are able to utilize a number line to prove their response to the first question. The third question gives students a chance to use area model to identify three-fifths of a 5-by-5 grid. The fourth question is a circle where students are expected to identify five-sixths of the area in the circle. The fifth question is a set of 36 stars, real-world objects, where students are expected to identify one-fourth of the total number of stars. The sixth question is two rectangular boxes that students are expected to use (as in measurement) to determine which quantity is greater two-fifths or three-eighths.
The assessment was administered during class for students in both classes on the same day. Participants in the 7th grade were given the assessment as a warm-up activity. Participants in the 9th grade were given the assessment after completing a test. In both classes, students were allowed 20 minutes to complete the five questions. During the assessment, students were read the directions if requested, individually. Hints, clues, suggestions, and solutions were not provided.
The research questions were analyzed by scoring the tests as correct, incorrect, or not attempted. Each item was individually reviewed and compared with the other participant's responses as well. The responses were analyzed to determine how students represent fractions and problem solve. In addition, analysis focused on student understanding of fractional knowledge.
Question one required students to determine the greater of two fractions (See Figure 1). Results indicated twenty-three correct responses, one non response, and an incorrect response. The student that did not respond was in the 7th grade. The student that provided the incorrect response was in the 9th grade.
In question two, students were asked to approximate the location of two fractions on a number line (See Figure 2). Four students were able to approximate the location of the quantities of and correctly, sixteen were inaccurate, and five students provided no response for the question. Surprising, there were four responses from students in both grades that resembled the model created in Figure 1. Figure 3 is an example of a correct model of the two fractional or rational quantities. Responses similar to that found in Figure 2 and 4 raise concerns in the analysis of students' understanding of comparing rational number quantities using a number line because of the assignment of values that differ from conventional rules of numeracy. Problem two in Figure 1 and 2 demonstrates that the student is using whole number reasoning and placing the fractions on the number line according to the magnitude of the denominators (Petit, Laird, & Marsden, 2010).
The image in Figure A was found on the answer sheet of a ninth grade student.
The image in Figure B was found on the answer sheet of a ninth grade student.
The image in Figure A was found on the answer sheet of a seventh grade student.
The image in Figure A was found on the answer sheet of a ninth grade student.
For question three, students were required to represent part of a whole using a grid. Nineteen responses were correct, two students did not respond, and five student responses were inaccurate. Of the two students that did not respond, one student was in the 7th grade, another student was in the 9th grade. Students in both grades provided an incorrect representation of the portion of the grid (See Figure 3). The questions that were correct all involve students shading in 15 of the 25 total squares or boxes on the 5-by-5 grid. One of the correct responses, Figure 5, shows how the student rationalizes the coloring of 15 boxes.
The image in Figure 5 was found on the answer sheet of a seventh grade student.
The image in Figure 6 was found on the answer sheet of a ninth grade student.
Question four required students to represent 5/6 using a circle model. Two responses were correct in representing 6 seemingly congruent sections of the circle (See Figure 7). All other students were not able to either draw the sections of equal size or approximated the shaded, five-sixths, of the circle. One of the correct responses was from a 7th grade student; the other correct response came from a 9th grade student. Some of the more interesting responses or frequent errors noticed are provided in the table below. Figure 7 shows that the 7th grader can approximate the value of five-sixths, but is not able to show the reason exact or approximate area confidently as seen in Figure 8. Figure 9 is very close to an accurate depiction of the area, however, the sections of the circle are not equivalent - there are four eighths and two fourths shown. Figure 10 is from the same student who provided rational in Figure 6; however, the student makes the assumption that the units of the circle as drawn similar to a grid are the same size.
The image in Figure 7 was found on the answer sheet of a seventh grade student.
The image in Figure 8 was found on the answer sheet of a ninth grade student.
The image in Figure 9 was found on the answer sheet of a seventh grade student.
The image in Figure 10 was found on the answer sheet of a ninth grade student.
Students were asked to identify ¼ of the total objects represented in question 5. Twenty-three students provided the correct response and two were incorrect. Although there were different methods used by the students who identified the fourth of all 36 stars, every student identified nine as being the fourth using real-world objects or tangible items. Of the two students that scored incorrectly on the problem, one circled all of the objects and the other circled only five.
Question 6 measured whether students were able to correctly determine if 2/5 is greater than 3/8 using bars. There were five correct responses and fourteen incorrect responses, demonstrating students are unable to represent the different rational number quantities as shaded regions of a set of equal sized sections. Six responses were blank. Of the five responses that were correct, three were from students in the 7th grade, two were from 9th graders. Figure 11 shows that the student understood to partition the rectangles into parts determined by the denominator of both fractions, but were unable to create equal sized-partitions, an error identified by Petit, Laird, Marsden (2010). Figure 12 seems to show the same mistake, however, there is not a clear cut response as to which quantity the student thinks is greater. Figure 13 represents a correct response with an appropriate model of proving the students reasoning and rationale.
The image in Figure 11 was found on the answer sheet of a ninth grade student.
The image in Figure 12 was found on the answer sheet of a ninth grade student.
The image in Figure 13 was found on the answer sheet of a seventh grade student.
The image in Figure 14 was found on the answer sheet of a seventh grade student.
Results demonstrated that students have difficulty representing fractions at the 7th and 9th grade level. Students demonstrated difficulty with proportional thinking throughout many of the tasks. While some students were able to successfully place the fractions on the number line in sequential order, they were not proportional. This was also evident when students used the area models to determine which fraction was greater and when they were asked to shade a fractional part of the circle model. According to Petit, Laird, & Marsden (2010), inaccuracy of models can be the result of having an incomplete knowledge about the importance of wholes being the same when compared to fractions.
Results did not indicate a distinct difference in fractional problem solving using representations between seventh and ninth graders. Most seventh and ninth grade students are able to use representations when solving fractions. However, results indicate that some students demonstrate a lack of understanding of fractional knowledge. This is particularly concerning, given that students begin working on fractions in the first grade and should begin to develop an understanding of fractions in the third grade. Students who are unable to use fractions to represent numbers equal to, less than, and greater than one and solve problems that involve comparing fractions by using visual fraction models and strategies lack a conceptual understanding of fractions that will continue to hinder their mathematical abilities as they are introduced to more complex math skills.