This is a paper I wrote for Research Design during Summer 2020. It is a proposal but not something that will be directly acted upon for research. It was completed for practice but does represent really good concepts.
Conceptualizing Attitudes Toward Mathematics and Exploring Racial Disparities after a Formative Assessment Intervention
Stephanie White
July 7, 2020
Introduction
Disparities in mathematics achievement have continuously raised concerns about why American students do not perform at the level of students from other countries. And, in the United States, there is an increased urgency to focus on teaching practices that impact access and equity in mathematics (National Council of Teachers of, 2014). In order to change results in student learning outcomes, teachers have to reimagine how they teach so that all students can have positive attitudes toward mathematic and learn with deeper understanding and success.
Attitudes toward Mathematics
Neale (1969) defined attitudes toward mathematics in a multi-faceted way as “a liking or disliking of mathematics, a tendency to engage in or avoid mathematical activities, a belief that one is good or bad at mathematics, and a belief that mathematics is useful or useless” (p. 632). Schoenfeld (2016) preferred to use the term ‘beliefs’ instead of attitudes and defined it as an “individual’s understandings and feelings that shape the ways that the individual conceptualizes and engages in mathematical behavior” (p. 26). Both of these terms refer to how a student may engage in mathematics and student feelings about mathematics. Often, researchers interchange ‘attitudes’ and ‘beliefs’ in their studies.
This proposal is grounded in the concepts of mathematical cognition (Schoenfeld, 2016) in which he used the term ‘belief’ instead of ‘attitudes.’ According to Schoenfeld (2016), mathematical cognition has five components: knowledge base, problem-solving strategies, monitoring and control, beliefs and affects, and practices. For the purpose of this study, I will focus on the Schoenfeld’s beliefs portion of his framework. The mathematical cognition framework proposes that beliefs influence mathematical engagement and how students conceptualize themselves as mathematics learners. Student experiences shape beliefs and those beliefs, in turn, influence how students do mathematics; often negative beliefs leading to negative outcomes. A core element of the mathematical cognition framework is that “whether acknowledged or not, whether conscious or not, beliefs shape mathematical behavior” (Schoenfeld, 2016, p. 28). This framework illustrates the importance of further understanding how attitudes toward mathematics interact with each other because of their impact on students’ willingness to engage in mathematics learning. Attitudes can be significant barriers to learning for some students in mathematics classrooms.
Looking at attitudes holistically, as shown in a Turkish study with 5th grade students focused on metaphors of beliefs, researchers found that individual students held both positive and negative attitudes toward mathematics (Çetinkaya, Özgören, Orakci, & Özdemir, 2018). Further, their study revealed that the most prominent student belief about mathematics was its difficulty and that negative attitudes toward mathematics increased as children progressed in grades. Other research has attempted to deconstruct the idea of attitudes into singular beliefs. Sahin (2018) researched the effects of specific attitudes toward mathematics using Trends in International Mathematics and Science Study (TIMSS) 2015 data and found that confidence was significantly related to better mathematics performance and had the greatest effect size (.207). Further supporting the impacts of students’ self-confidence in mathematics, Di Lonardo Burr (2020) focused on confidence (defined as affect and perceived self-efficacy) and found relationships between confidence and mathematics performance. Another facet of attitudes toward mathematics is the idea of student mindsets; the idea that intelligence is malleable or not. Mindsets have been shown to influence mathematics achievement more than other subject domains (Jonsson & Beach, 2012). In a longitudinal study of the effects of mindset-related training and interventions in middle school classrooms, Anderson, Boaler, and Diekmann (2018) showed that students with teachers who participated in the study and used mathematical mindsets teaching interventions performed significantly higher on state standardized tests in mathematics than those students whose teachers did not adjust their teaching practices. Research conducted by Degol et al. (2018) supported the connection between students’ mindsets and mathematics performance. There exists an array of definitions and constructs of attitudes toward mathematics and how they influence performance or other student outcomes. It is important to consider various types of attitudes toward mathematics and how they interact with one another since it is apparent that students have multiple attitudes toward mathematics working together as they are learning. The ability to better define attitudes toward mathematics universally becomes more essential to understanding how to increase positive attitudes and potentially impact learning. Additional research is needed to better conceptualize attitudes toward math into universally accepted definitions.
Formative Assessment as an Equity Practice
To better understand why there are student differences in access and equity in mathematics education, defining attitudes toward mathematics can be used to explore if there are differentially negative attitudes among traditionally underserved student populations. Equity occurs when we can no longer predict outcomes based on student characteristics and when every student has the support they need to excel (National Council of Teachers of, 2014). Student differences are strengths to be utilized in the classroom but if negative attitudes toward mathematics exist differentially in certain student populations, those attitudes may create barriers toward access and equity in mathematics since attitudes can impact learning. Formative assessments, those for and throughout learning instead of cumulative assessment of learning, can foster equity and create more dynamic environments in mathematics classrooms (Duckor, Holmberg, & Becker, 2017). Teaching through formative assessment involves students at higher levels and elicits student thinking so teachers can facilitate learning transparently. For this study, a formative assessment intervention was selected because previous studies have shown that its use led to powerful impacts on achievement and dramatic increases in student learning (Duckor et al., 2017; Wiliam & Leahy, 2016).
Focus on the Transition to Middle School
Middle school is particularly interesting for this research question because of characteristics that are unique to this period for students. A prominently researched issue is the transition to middle school, when students often falter academically. Blackwell (2007) highlighted several reasons why middle school students have distinct challenges: maturation, shifts in academic expectations, social demands, increased competition, increased self-focus, and disengagement from school. In middle school, students start to experience social stress, stress that comes from perceived differences in abilities, and “stereotype threats” (Good, Aronson & Inzlicht, 2003). Specifically, for mathematics, there are increased inequities for students in middle schools (Berry, 2008). Given students’ unique needs in middle school, focusing on the impact of formative assessment on improving attitudes toward mathematics for students entering middle school may help educators mitigate the issues facing middle school students in mathematics.
Research Design
This study seeks to explore the relationships between survey items related to students’ attitudes toward mathematics in 4th and 5th grade within a large, urban school district to determine if the components of attitudes toward mathematics that exists for such a population of students is the same as a previously determined structure (Mathematical Self-Perceptions, Enjoyment of Mathematics, Perceived Usefulness of Mathematics). After confirming the factor structure for attitudes toward mathematics for the study sample and after the year-long experiment focusing on formative assessment, I will examine race disparities among the attitudes toward mathematics factors. The goal of this study is to add to existing research aimed at better understanding attitudes toward mathematics and how formative assessment might lead to a greater increase in positive attitudes toward mathematics for non-white students as they enter middle school.
Research Questions
The main hypotheses for this study are that there are three components of attitudes toward mathematics and that non-white (including Hispanic) students will have more negative attitudes toward mathematics than their white (not including Hispanic) student counterparts. Further, formative assessment teaching methods will differentially improve attitudes toward mathematics for non-white students than white students. Two research questions can be answered from this study.
Research question 1: What are the components of students’ attitudes toward mathematics as they exit elementary school and how are they structured?
Research question 2: Considering the construct of attitudes toward mathematics, are their race/ethnicity differences on the identified components after the intervention?
To answer the first research question, I will employ Confirmatory Factor Analysis (CFA) using participant pretest results from the Math and Me Survey (M&MS) to confirm that the sample has the same components as hypothesized. For the second research question, I will conduct a randomized control trial in which classrooms in participating schools will be randomly assigned to the formative assessment intervention or to the control condition, standard teaching as usual, for a year of instruction with or without the intervention. A posttest M&MS survey will be administered to investigate potential differential shifts in attitudes toward mathematics at the end of the school year.
Participants and Setting
The transition to middle school is when students often struggle academically (Blackwell Dweck & Trzesniewski, 2007). Since the M&MS has been used for elementary students, for this study, I propose using this instrument to survey 4th and 5th grade students in Jefferson County Public Schools (JCPS) before they enter middle school. I will work with JCPS regarding which elementary schools will participate in the study such that students in those grades at those schools are representative of the district’s student 4th and 5th grade population on demographics such as race/ethnicity, socio-economic status, English learner status, special needs, mathematics ability and the district’s teacher population on years of experience. It will also be required that for schools to be included in random assignment to conditions, no new mathematics curriculum be launched during the year of the intervention study to help avoid interaction with another intervention.
In partnership with JCPS with students’ guardian approval, and after university internal review board approval, students from 4th and 5th grade will take part in this survey before intervention implementation and again at the end of the school year after intervention. From the randomly assigned classroom, random samples of at least 150 students from each condition will be chosen for data analysis. The goal for random sampling will be at least 300 students since there are 30 items on the survey and it is recommended to have 5 to 10 participants per item (Yong & Pearce, 2013, p. 80). The study will take place over a school year to minimize attrition between school years. Still, I will track attrition by condition throughout the study as well as overall attrition in a CONSORT flow diagram and look for differential attrition rates in student characteristics. If differential attrition rates emerge by treatment group, I will analyze baseline differences in the effect sizes on the M&MS pretest to determine if the attrition caused an imbalance in the baseline measures.
Intervention
The goal of formative assessment is not just checking for understanding but developing teacher actions that will involve students and elicit student thinking so teachers can continuously facilitate learning. Teaching using formative assessment is not inherent in all teaching programs and requires training in order to be implemented well. Teachers will be trained in the key aspects of formative assessment (William, 2007).
1) Identifying where learners are.
2) Identifying the goal for the learners.
3) Identifying paths to reach the goal.
Teacher training will also involve providing teachers with teacher actions they can take to regularly incorporate formative assessment in their classrooms. These teacher actions will be based on the six effective teacher actions for improving mathematics achievement: making expectations explicit (participation and mathematical work), coaching students (teacher intervention), attending to students’ local context (connection to cultural or social context of student), attending to language (revoicing to appreciate student contributions), attributing mathematical authority to students (time for student reflection, discussion), and attending to classroom community (explicit action taken to build a productive environment) (Duckor, Holmber & Becker, 2017).
During the first month of the school year, teachers will undergo two days of training. Throughout the school year, during implementation, teachers in the intervention condition will received two additional days of training and support in the fall and in the spring. Also, mathematics coaches within the school who have also been through the training will be available to assist teachers from the intervention condition within their classrooms throughout the year. These coaches can also help monitor classrooms for implementation fidelity.
Data Analysis
For this proposal, I suggesting using the M&MS because it has been widely cited and had high internal and external validity (Adelson & McCoach, 2011, pp. 239-240). The M&MS confirmatory factor analysis has been conducted on a nationwide sample and again on a sample of schools in different types of areas (rural vs. urban). Additionally, this study focuses on elementary students, therefore the M&MS is a reasonable instrument to use. The M&MS has identified three components of attitudes toward mathematics: Mathematical Self-Perceptions, Enjoyment of Mathematics, Perceived Usefulness of Mathematics. Given the unique characteristics of a large urban district, I will first confirm these components for the sample before using the factors as dependent variables.
Next, after checking model assumptions, I will use a two-level multivariate hierarchical linear model (HLM) since students are nested within classrooms within schools and there will be three outcome measures, the factors from the survey. This model will also help mitigate possible issues due to attrition since multivariate HLM is the best way to deal with missing data when the data is nested (Goldstein, 2003, p. 6). Through this model, I will test the impacts of the formative assessment intervention (with binary coding) and race (white and non-white) on the survey measures from students at the end of the school year in the three components of attitudes toward mathematics. The HLM model will have controls for the M&MS pretest scores on the three outcomes and controls for the student characteristics that were checked upon randomized assignment. I will use an intention-to-treat framework to analyze measures based on the group to which students and teachers were initially assigned. The results will be presented as impact estimates expressed in effect sizes testing significance at the p < .05 level.
Potential Limitations
This experiment has some potential limitations. Since this is a teacher-level intervention, but student-level measures, implementation fidelity may impact the results. Though teachers will be trained and supported as well as monitored by mathematic coaches, there will be variability in the extent to which teachers modify their teaching styles. Also, because the pretest is a survey of attitudes, both the teachers and students may presume the purpose of the study. Such presumptions may influence their behaviors toward positive attitude answers in the post-intervention survey because they feel compelled to respond according to the presumed impact of the experiment. Teachers’ presumptions may change other ways they teach. For example, if a teacher presumes that the goal of the formative assessment intervention is to increase positive attitudes toward mathematics, the teacher may adjust language, testing, or other teaching strategies that may impact the results.
References
Adelson, J. L., & McCoach, D. B. (2011). Development and Psychometric Properties of the Math and Me Survey: Measuring Third through Sixth Graders’ Attitudes toward Mathematics. Measurement and Evaluation in Counseling and Development, 44(4), 225-247. https://doi.org/10.1177/0748175611418522
Anderson, R. K., Boaler, J., & Dieckmann, J. A. (2018). Achieving elusive teacher change through challenging myths about learning: a blended approach. Education Sciences, 8(98). https://doi.org/10.3390/educsci8030098
Berry, R. Q. (2008). Access to Upper-Level Mathematics: The Stories of Successful African American Middle School Boys. Journal for Research in Mathematics Education, 39(5), 464-488.
Blackwell, L., S., Dweck, C. S., & Trzesniewski, K. H. (2007). Implicit Theories of Intelligence Predict Achievement across an Adolescent Transition: A Longitudinal Study and an Intervention. Child Development, 78(1), 246-263. https://doi.org/10.1111/j.1467-8624.2007.00995.x
Çetinkaya, M., Özgören, Ç., Orakci, S., & Özdemir, M. Ç. (2018). Metaphorical perceptions of middle school students towards sath. International Journal of Instruction, 11(3), 31-44. https://doi.org/10.12973/iji.2018.1133a
Degol, J. L., Wang, M. T., Zhang, Y., & Allerton, J. (2018). Do growth mindsets in math benefit females? Identifying pathways between gender, mindset, and motivation. Journal of youth and adolescence, 47(5), 976-990. https://doi.org/10.1007/s10964-017-0739-8
Di Lonardo Burr, S. M., & LeFevre, J.-A. (2020). Confidence is key: unlocking the relations between ADHD symptoms and math performance. Learning and Individual Differences, 77. https://doi.org/10.1016/j.lindif.2019.101808
Duckor, B., Holmberg, C., & Becker, J. R. (2017). Making Moves: Formative Assessment in Mathematics. Mathematics Teaching in the Middle School, 22(6), 334-342. https://doi.org/10.5951/mathteacmiddscho.22.6.0334
Goldstein, H. (2003). Multilevel Statistical Methods (3rd ed.). Arnold.
Good, C., Aronson, J., & Inzlicht, M. (2003). Improving adolescents’ standardized test performance: An intervention to reduce the effects of stereotype threat. Journal of Applied Developmental Psychology, 24(6), 645-662. https://www.doi.org/10.1016/j.appdev.2003.09.002
Jonsson, A.-C., & Beach, D. (2012). Predicting the use of praise among pre-service teachers: the influence of implicit theories of intelligence, social comparison and stereotype acceptance. Education Inquiry, 3(2), 259-281. https://doi.org/10.3402/edui.v3i2.22033
National Council of Teachers of, M. (2014). Principles to Actions: Ensuring Mathematical Success for All. National Council of Teachers of Mathematics.
Neale, D. C. (1969). The role of attitudes in learning mathematics. The Arithmetic Teacher, 16(8), 631-640.
Sahin, M. G., & Öztürk, N. B. (2018). How classroom assessment affects science and mathematics achievement? Findings from TIMSS 2015. International Electronic Journal of Elementary Education, 10(5), 559-569. https://doi.org/10.26822/iejee.2018541305
Schoenfeld, A. H. (2016). Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics (Reprint). Journal of Education, 196(2), 1-38. https://doi.org/10.1177/002205741619600202
Wiliam, D., & Leahy, S. n. (2016). Embedding Formative Assessment. Hawker Brownlow Education.
Yong, A. G., & Pearce, S. (2013) A Beginner’s Guide to Factor Analysis: Focusing on
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