Running head: How large denominators are leading to large errors 1

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1 Running head: How large denominators are leading to large errors 1 How large denominators are leading to large errors Nathan Thomas Kent State University

2 How large denominators are leading to large errors 2 Abstract When it comes to fractions and working with them, they can be difficult to understand. Through our research, we attempted to bring a greater understanding to how we as individuals learn fractions. With this, we can move forward to teach fractions in a way that lets us have a better understanding of fraction magnitudes and how to work with them. For our study, we tested three hypotheses. First, we hypothesized that it would be more difficult to determine the magnitude of a fraction when there is a large denominator involved. Coding our data, we also hypothesized that it would be more difficult to determine the larger of two fractions when there was a smaller decimal distance between them than when there was a larger one. Finally, we tested for a timing element, expecting to see more time taken for the larger denominator fractions as well as the comparison tasks having a smaller decimal distance. We used a magnitude comparison task in which participants would determine the larger of two fractions, ranging in a variety of different decimal distances. We found support for our hypothesis that it is more difficult to complete a fraction containing a large denominator, as well as our hypothesis that participants perform better on magnitude comparison tasks when there is a greater decimal distance between the two numbers. Through this information, we hope to be able to change our teaching methods in order to allow students to have a better understanding of fractions.

3 How large denominators are leading to large errors 3 Introduction Determining and expressing the magnitude of fractions is difficult for all persons due to the fact that they are not taught in a way that is allowing individuals to code it for long-term knowledge, depriving students to the point that that the reader of this paper, we would guess, has had a difficult time gaining a good understanding of fractions. Fractions are difficult to understand no matter if you come from a high social economic status versus a low one, a private school versus a public, or any education system, for that matter. In today s education system, students are not receiving the topic of fractions in a way that is allowing them to gain a good understanding of how to solve these types of mathematic problems. Students are being presented the material for fractions, but this does not mean that the information is sticking with the students. No matter the age of a person, determining the value of a fraction is difficult. Due to the fact that our education system is spending so much time on fractions, and because it is important that students be able to recall what they learn later on in life, this problem needs to be fixed. Fractions are not being presented in a way that allows students to gain a good understanding of working with fractions. The reason the material is not being presented correctly is because we are still unsure of a way to present the material that will allow students to learn the information they need regarding fractions and retain it. Through this research, we hope to find information that will allow us to gain a better understanding of mathematical cognition with fractions, and determine the best presentation method of the material so that students can be taught in a way that will help them to grasp a better understanding of fraction magnitudes. A magnitude comparison task is when an individual attempts to compare two fractions, determine which of them has the large or smaller decimal value, then report on which fraction

4 How large denominators are leading to large errors 4 they believe to have the greater or lesser decimal value. Through this introduction, we will show you support for how fraction magnitude is determined by an individual, through the way an individual solves certain magnitude comparison tasks in fractions. We will look at what past researchers have found and the reasons why their work is relevant to our study. Solving a fraction is difficult because the magnitude of the fraction is confusing based on the singular parts of the fraction. Meert et al. (2010) explains this in their study, in which they discuss their theory of how fractions components in a magnitude comparison task can impact how easy or difficult it is to choose the larger fraction. In their study, they explain that when a fraction has similar parts, such as equivalent numerators or equivalent denominators, solving the fraction takes less time and a lower understanding of the fractions is needed in order to determine which fraction is larger. When comparing fractions that do not have a common component, it requires the participants to gain a better understanding of the fraction holistically, and determine the true magnitude of the fractions in question. This study enhances the idea that looking at a fraction holistically can help us gain a better understanding of the magnitude of it, and it is more difficult to gain a magnitude understanding of a fraction when it has a large denominator. In the numerical distance effect, it is hypothesized that when an individual has to compare two fractions that are closer in distance than fractions that are further in distance, there is a much greater chance that more mistakes and errors will be made than with those that have a greater decimal distance between the numbers that are being compared. In the global processing strategy, comparisons of the magnitudes will be based on an estimated value of both of the fractions involved. The global processing strategy is the opposite of the segmented processing strategy, in which an individual will compare the numerator of one fraction to the numerator of another fraction, and the denominator of one with the denominator of the other fraction, and

5 How large denominators are leading to large errors 5 determine the magnitudes based on this information. Ishbeck, Schocke and Delazer (2009) studied reaction to a comparison magnitude task alongside an fmri study. They hypothesized that there may be a global or segmented processing strategy in whether or not an individual solved the fraction based on all of its parts, as a whole, or rather, compared one fraction to the other based on the numerator and denominator parts that make it up. In another study with adult participants, DeWolf & Vosnaidou (2015) found that adults fall for the whole number bias as well. The whole number bias is evaluating the value of a fraction based on its whole number parts, such as pulling out the 2 and 3 in 2/3 and using them to try to understand the value of the fraction as compared to looking at it as a division problem and seeing it where 2 is divided by 3 and we see a value of ~.6667 rather than a whole number. The whole number bias is similar to and includes the segmented processing strategy because individuals are segmenting the individual parts of the fraction rather than looking at it holistically. When adults see the whole number parts of the fractions, this leads them to compare the magnitudes of four and three, when comparing fractions 1/3 & ¼, rather than comparing the fraction magnitude as a whole, in which they would understand that 1/3 is greater than ¼. In their study, DeWolf & Vosnaidou (2015) manipulated the distance between the fraction magnitudes. They found that the closer together two fractions were, the smaller the distance between their magnitudes, the more likely it was that an adult would look to the whole number parts of the fraction in order to try to determine magnitude, rather than trying to determine the true magnitude of the fraction in question. Obersteiner et al. (2013) explains that the size of a number, whether in a numerator or a denominator, is likely to cause confusion and mistakes when discovering fraction magnitude. With response time as a factor, Obersteiner et al. (2013) found that even with expert

6 How large denominators are leading to large errors 6 mathematicians, they had a faster response time on problems that had common numerators or denominators than for problems that had differing parts. Finally, another study in which response time was measured was a study conducted by Campbell & Hyrenyk (2012). They used whole numbers to measure how quickly it took participants to solve simple math problems. They used a scale and had large (bigger than 25) and small numbers and compared the response time of participants based on this information. Their study also had a component in which the participant would respond with how the participant came up with their answer after they determined what they believed the answer to be. Their data supported the idea that larger math problems are more difficult to solve than problems that have smaller individual parts. The present study Our belief is that when the denominator is manipulated, specifically to encompass a large denominator, magnitude comparison is even more difficult to complete with accuracy. We hypothesized that when the denominator is manipulated, larger denominator fractions are harder and take more time to solve in a magnitude comparison task than fractions that are made up of small denominator parts, and that it would be more difficult for participants to determine the value of a fraction when there was a smaller decimal distance between the two fractions being compared. This is due to the fact that it is harder for individuals to have a true understanding of the magnitude of the fractions in question, and they are less likely to make an accurate estimation of the true size of the fraction. We controlled for decimal distance between fractions and expected to see a greater number of mistakes when a smaller decimal distance was present between the two fractions. We suspected that our participants would have a more difficult time solving fractions that were closer together (have a difference of.1 or.3) than solving a magnitude comparison task in which

7 How large denominators are leading to large errors 7 the difference between the fractions is greater (.7 or.9). We used fraction differences of.1,.3,.5,.7, and.9. This allowed us to gain a good understanding of whether or not the numerical distance effect had support. In our study, participants were to choose the fraction with the larger magnitude and then participants would explain their rational for this choice as a part of another study. Participants followed a link to Qualtrics, a research gathering site that would collect participant s responses. Our reason for this study was to determine that large fractions are in fact more difficult to solve. Our research will add to the literature that is already being studied because it will allow researchers to have more support for the fact that large fractions are more difficult to look at holistically and gain an understanding of their magnitude. Method Participants For our study, we had 52 participants. Participants were gathered by students in a cognitive psychology lab at Kent State University, each of whom contacted five people, to take the exam. Participants also were able to log in through a system and take the exam when the number of participants was not close to what we as researchers expected and hoped for. Participants were also Kent State University students. Participants could have been chosen at random or acquaintances of the class members. Participant addresses were collected to make sure there was not any overlap. Participants that logged in received extra credit points in a psychology class of their choosing through a points system managed by the Department of Psychology. The average age of participants was years with a standard deviation of 1.72, 75.51% of which were white or Caucasian, 8.16% black, 8.16% Asian, 6.12% Hispanic, and

8 How large denominators are leading to large errors % Nigerian-American. There were 11 males, 36 females, one agender, and one prefer not to respond. There was no specialization or known advanced math skills held by any of the participants. We removed the data of three participants because they did not respond to our particular section of questions on the survey. Procedure When participants opened the test, they were prompted to sign the informed consent form before they were able to take part in the experiment. The informed consent let participants know that they could at any time leave the study, and that their identity would be kept confidential either way. We also explained that if they for some reason wanted their data removed, they could contact us and have their responses removed from our data. Participants were students at Kent State University. They could have been anywhere when they took the exam as long as they were near an internet source, but we do not know the specifics of the environment for our participants. The methodology we used to test our hypothesis was administering a Qualtrics examination in which participants attempted to choose the larger of two fractions. This exam was an online exam that participants logged in to take, and their responses were recorded after completion. Participants were to choose the greater magnitude of two fractions. There were a total of 20 magnitude comparison tasks. We used ten fraction comparisons that had denominators less than ten, and ten fraction comparisons that had denominators greater than ten. In each of the comparison types (small and large denominator), we paired fractions by varying distances in fraction magnitude. For two of the fractions in each group, they varied by distances of 0.1, 0.3, 0.5, 0.7, and 0.9. The order of magnitude differences were mixed and distributed

9 How large denominators are leading to large errors 9 randomly. The type of fraction, large or small denominator, was also random, and participants randomly solved large and small denominator fractions. The reason this task was appropriate to measure what we desired was that it gave participants differing fractions, all with different denominators. While some denominators were considered small (less than ten) and other denominators were considered big (larger than ten) each of the fractions in one type of task had other fractions that were similar or equal in distance in the other group. While the fractions being compared were different, the fact that each comparison had a large or small denominator fraction that was the same distance gave reliability to our experiment. In our directions, we explained that when we said we wanted the larger of the fractions, we wanted the participant to choose the fraction that was closest to the value of one, as all of our fractions were less than one. We also explained that response time was important as well, and asked participants to focus on the task until completed and take it in a quiet environment. In addition, we asked that our participants not use a calculator or the help of anyone other than themselves to choose the larger magnitude fraction. Tasks/Measures For our data, we used a magnitude comparison task (Fazio et al., 2014). Participants were to look at two different fractions and choose the one that had a greater magnitude. We also judged participant response time. The fractions and comparisons used in our experiment can be found in Appendix 1. We coded our data by accuracy of response and length of time it took for participants to respond. We judged whether or not fractions with larger denominators (a denominator greater

10 How large denominators are leading to large errors 10 than ten) are more difficult to solve than fractions that have a denominator less than ten. We used a t-test to test our data. This was because we were comparing the answers of response time and accuracy for the large denominator fraction and the small denominator fraction. Results With our data analysis, we tested our hypothesis that there would be a larger number of errors on problems having a large denominator compared to problems containing a smaller denominator part. The problems were manipulated for decimal distance, having two large and two small fractions with a decimal distance of.1,.3,.5,.7, and.9. We tested to see if there was any significant difference between the problems at different distances. This was to test our hypothesis that there would be a larger number of errors when trying to determine the larger of two fractions when the decimal distance between them was smaller, such as a distance of.1 or.3. We also tested the time it took participants to respond to each question, hypothesizing that it would take longer for participants to complete the problems that had a larger denominator. For our data, we used JASP to determine if we had significant results or not. We did a paired samples t-test as well as an ANOVA. We took the percent of problems accurately completed for the small denominator fraction problems and compared it to the data from the large denominator fraction problems. Our statistics showed support for our hypothesis, giving us significant results, t(48) = 3.808, p <.001, d= Small denominator level - M=91.9% correct, SD=.148 large denominator level - M=76.5% SD=.340. This shows support for our hypothesis that fractions with larger denominators are harder to solve, as the accuracy at the large denominator level was significantly lower overall.

11 How large denominators are leading to large errors 11 We also did a paired samples t-test in controlling for response time. Our statistics were not significant, p=.996, t(48)=.005, d=7.347e-4. We found the median response time it took participants to respond to each question. Small denominator response times M= seconds, SD 4.997, and for the large denominator response times - M=5.799 seconds SD= When we look at the data, we see that the median response time is very similar for both large and small denominator fractions. As seen in Table 3, we ran a repeated measures ANOVA testing the decimal distance for large and small denominator fractions. We found significant data suggesting that it is more difficult to determine the larger of two fractions when you are working with fractions that are similar and closer in decimal distance, F(4,192) = , p<.001. Discussion The hypotheses that we tested were: larger denominator fractions would be harder to solve than smaller denominator fractions, that it would take longer to solve a magnitude comparison task between two large denominator fractions than when comparing two small denominator fractions, and that fractions that had a smaller decimal distance would be harder to solve than fractions that had a larger decimal distance. Our hypotheses were supported by the existing literature for the hypothesis that a smaller decimal distance makes magnitude comparison tasks harder to solve, and also supported our hypothesis that a larger denominator makes a fraction problem more difficult to solve than when a small denominator is present. These hypotheses were not only supported by the existing literature on the topic but also by the data we received from our experiment. Our hypothesis that larger denominator fractions take more time to determine fraction magnitude than determining the magnitude of a fraction that

12 How large denominators are leading to large errors 12 contains a small denominator part was not supported, but we believe this was due to a lack of control on the environment of subjects as they took our survey. In order to study our hypothesis, we used a magnitude comparison task in which participants would choose between two fractions to determine which fraction was larger (closer to one on the number line). We controlled the experiment by manipulating the decimal distance between the fractions. Each participant received ten comparison tasks that had a large denominator (between 11 and 20) and a small denominator (between 2 and 10) that had two problems with a decimal distance of.1,.3,.5,.7, and.9. each. This made a total of four problems with a distance of.1, etc., two with large and two with small denominator fractions. We also timed the amount of time it took participants to make a decision, being the time when they would make their first click and decision on an answer. Our purpose for studying this was to gain a better understanding of what it is that causes individuals to learn fractions and what can be done to improve the way fractions are taught so that we can code fractions in a way that will make for better and easier recall later on. We also hoped to add to the existing literature while showing support for other hypotheses. Our hypothesis that larger denominator fractions are more difficult to solve was supported by the work previously done by Campbell & Hyrenyk (2012). In their previous research, they have looked at larger numbers in math problems and found that common small numbers led to more accuracy and a faster response time when it came to answering various types of math problems. Oberseiner et al. (2013) has also found that whether in the numerator or the denominator, when there are larger numbers involved it is more difficult to determine which is the larger of the two given fractions in a magnitude comparison task. While their research does not deal directly with fractions only, it does expand on the fact that when there are larger

13 How large denominators are leading to large errors 13 numbers involved, it is more difficult to solve that math problem than when there are more familiar numbers being used. More research that supports our hypothesis is that research done by Meert et al. (2010). When the parts of the fraction are manipulated, we see an adverse relationship when we begin to get outside the realm of what we would refer to as common numbers. With our study, we specifically manipulated the size of the denominator and took the individual denominator parts to a higher value, thus, using numbers that are not often used in everyday life. Because of this, we saw a similar pattern to the Meert et al. (2010) study. When the decimal distance between two fractions that are being compared in a magnitude comparison task is smaller, differing in distance near.1 or.3, it is more difficult to determine which of the fractions is the larger of the two fractions (DeWolf & Vosnaidou, 2015). When an individual is comparing two fractions that have a decimal distance of.7 or.9 between them, this is an easier task to complete due to the fact that the subject can determine which is larger or smaller based on an understanding of where a fraction is approximately on the number line, and can make their determination with a smaller amount of data knowledge, because they do not need to know the exact value of the fractions being compared. DeWolf & Vosnaidou s (2015) research supports our hypothesis that fractions that are closer together in decimal distance are more difficult to determine which is greater in a fraction magnitude test. Oberseiner et al. (2013) has completed research that supports our claim that it takes longer to determine the larger of two fractions when there is a large denominator present. They found that when the parts of a fraction are easily recognizable, faster response times are seen than when it is difficult to recognize the single parts of a fraction. While our study was constricted by the fact that we were not able to interact with our participants, we do believe that

14 How large denominators are leading to large errors 14 conducting this study in a laboratory setting would allow us to control the environment enough for us to get the results we were desiring. Some limitations to our study were our participants and those that we had access to. We were limited to college students, and could not get a very diverse population. We were limited by the type of study that we could perform. We needed to conduct the study through the use of Qualtrics, and this limited us in the way we could interact with those we were studying. We believe this may be the major limitation causing us to have insignificant results on our timing task. As we would not be interacting with participants, the timing was inconclusive as we could not measure participants in the lab, determine if anyone got distracted during the task, or the reason why they were able to come up with their answer as quickly or slowly as they did. Another limitation of our study was that we were not able to understand the thought process individuals went through in order to solve or pick the larger fraction, or if they used a calculator to help them solve the task. Our studies strengths, however, were that we did find support for the majority of our hypotheses. Through this study, we were able to find support for the idea that large denominator fractions are more difficult to solve, and that explaining and teaching fractions needs to be done using smaller number parts in order to allow students to gain a good understanding of the way they need to look at fractions in order to solve them. We also found support for our hypothesis that when completing a magnitude comparison task, it is more difficult to determine the larger of two fractions when they have a small decimal distance between them. This allows us to follow the understanding that when first teaching fractions, students will learn better and understand more if there is a larger decimal distance between the two fractions being compared. This support allows us to move forward understanding that these elements can make learning easier or

15 How large denominators are leading to large errors 15 harder, and thus manipulate our teaching for the better in order to give students a better understanding of how to solve fraction problems. In the future, a replication of this study would better be completed inside the laboratory. This would allow the experimenters to conduct their research in a way that they could study the time it takes participants to complete each problem. It would also give experimenters more control over the environment that each participant is taking the study in, making a quiet room where subjects can remain focused during their comparison tasks. This will allow for a controlled environment in which all participants will be completing the task in the same environment, giving the participants a chance to perform at the best they can, and thus giving researchers control across participants. Through this research, we have gained a greater understanding of the time it takes fractions to be processed and coded and learned by an individual. Fractions that have large number parts take longer to learn and/or understand, so starting out with smaller denominator fractions can help students grasp an understanding of the numerator and denominator before that student needs to complete difficult fraction problems containing large individual parts. If students can understand smaller fractions and get a good sense of their magnitude, then larger denominator fractions can begin to be used in teaching, as they will take time to grasp and gain a holistic understanding of their parts.

16 How large denominators are leading to large errors 16 Appendix 1 Table 1 T-Test Paired Samples T-Test t df p Cohen's d Percentage correct Small - Percentage Correct Large < Descriptives N Mean SD SE Percentage correct Small Percentage Correct Large

17 How large denominators are leading to large errors 17 Table 2 T-Test Paired Samples T-Test t df p Cohen's d Median Response Time Small - Median Response Large e -4 Descriptives N Mean SD SE Median Response Time Small Median Response Large Table 3 Repeated Measures ANOVA Within Subjects ANOVA Sum of Squares df Mean Square F p Decimal Distance ᵃ 4 ᵃ ᵃ ᵃ <.001 ᵃ Residual Denominator size Residual Decimal Distance Denominator size ᵃ 4 ᵃ ᵃ ᵃ ᵃ Residual Note. Type III Sum of Squares ᵃ Mauchly's test of sphericity indicates that the assumption of sphericity is violated (p <.05). Between Subjects ANOVA Sum of Squares df Mean Square F p Residual Note. Type III Sum of Squares

18 How large denominators are leading to large errors 18 References Campbell, J. I. D. & Hrenyk, J. (2012). Operand-operator compatibility in cognitive arithmetic. Canadian Journal of Experimental Psychology, 66, DeWolf, M., & Vosniadou, S. (2015). The representation of fraction magnitudes and the whole number bias reconsidered. Learning and Instruction, 37, Fazio, L. K., Bailey, D. H., Thompson, C. A., & Siegler, R. S. (2014). Relations of different types of numerical magnitude representations to each other and to mathematics achievement. Journal of Experimental Child Psychology, 53-72, 123. Ischebeck, A., Schocke, M., & Delazer, M. (2009). The processing and representation of fractions within the brain: An fmri investigation. NeruroImage, 47, Meert, G., Grégoire, J., & Noel, M. (2010). Comparing 5/7 and 2/9: Adults can do it by accessing the magnitude of the whole fractions. Acta Psychologica, 135, Obersteiner, A., Dooren, W. V., Hoof, J. V., & Verschaffel, L. (2013). The natural number bias and magnitude representation in fraction comparison by expert mathematicians. Learning and Instruction, 28,