Characteristics of Proficiency-Based Learning and Their Impacts on Math Anxiety in the Middle Grades

Abstract

Math anxiety can have an extraordinary impact on middle level students’ ability to learn. However, certain classroom practices may be able to mitigate this impact. In terms of assessment practices, proficiency-based learning, also called competency-based learning or standards-based grading, may help establish a learning environment that either prevents the development or decreases the influence of anxiety for middle level mathematics students. The present study analyzed the relationship between specific characteristics of proficiency-based learning and students’ anxiety about mathematics. Using a multiple case study design, we examined the practices in two middle level mathematics classrooms and how they related to the experiences of five students in those classes who self-identified as having math anxiety. We focused specifically on the practices of establishing clear learning objectives, implementing assessment for learning, providing effective feedback, promoting reflection, and providing opportunities for reassessment. Data collection included interviews of students and teachers, in-class artifacts, and class observations. We found that reassessment was the characteristic of proficiency-based learning most closely associated with alleviating students’ anxiety. We also found that math anxious students can benefit from effective feedback and clear learning objectives, particularly when presented simultaneously.

Keywords:

• assessment
• proficiency-based learning
• math anxiety
• middle level education
• mathematics teaching

Introduction

Perhaps more than in any other content area, students suffer from anxiety in mathematics. Mathematics anxiety, or math anxiety, is defined as an adverse emotional reaction to mathematics that can impact one’s ability to perform (Maloney & Beilock, 2012). While math anxiety can develop as early as age six, the middle grades appear to be a critical period for shaping students’ views of mathematics (Ma & Xu, 2004). Math anxiety can impact middle level students’ abilities to learn and can have lasting effects through high school (Field et al., 2019). Math anxiety is often associated with testing anxiety and general anxiety, however it is its own unique concept (Carey et al., 2017). While there have been several suggestions of ways to alleviate math anxiety, changing traditional assessment practices through cultivating an environment centered on learning has the potential to be particularly effective (Furner & Gonzalez-Dehass, 2011; Knight & Cooper, 2019).

Across the country, proficiency-based learning, also called competency-based learning or standards-based grading, is gaining widespread attention as a way to address shortcomings and inequities in traditional grading practices (Levine & Patrick, 2019). Proficiency-based learning refers to a system in which instruction targets a set of clearly defined, common learning expectations, assessments directly align with these expectations, and students are able to reassess without penalty as their learning progresses (Great Schools Partnership, 2016). In Vermont, schools across the state are rapidly adopting proficiency-based learning in response to the passage of Act 77. This legislation mandates flexible pathways to graduation and proficiency-based graduation requirements in grades 7–12 (Vermont Agency of Education, 2017, n.d.). The implementation of proficiency-based learning can vary substantially from school to school and even classroom to classroom. However, there are some clearly identified best practices of proficiency-based learning that all systems have in common (Great Schools Partnership, 2016).

Because students in a proficiency-based system have multiple opportunities to demonstrate mastery and can allow their learning to develop at their own pace, it is possible that the fear of failure, a major component of math anxiety, could be reduced (Martinez, 1987). However, there is little research on the relationship between proficiency-based learning and math anxiety. As teachers in Vermont and across the country plan and revise their implementation of proficiency-based learning, awareness of its impact on students who are anxious about math is essential. Understanding this interaction at the middle level is of particular importance, because during the middle grades students tend to develop more negative attitudes toward school, and specifically about math (Falco, 2019; Midgley et al., 1989; Raphael & Burke, 2012).

In this study, we used a multiple case study approach to examine practices in two middle level mathematics classes in Vermont using proficiency-based learning and their relationship to the math anxiety of five self-identified, math anxious students. Specifically, our study addressed the question: How do specific characteristics of proficiency-based learning interact with students’ math anxiety?

Background

Math Anxiety

Math anxiety is “a feeling of tension that inhibits a person’s ability to do math” (Furner & Marinas, 2016, p. 26). This tension can be incredibly debilitating when learning mathematics. Adverse emotional reactions, such as being overly nervous, nauseous, or developing a migraine can spark from just opening a textbook (Maloney & Beilock, 2012). Math anxiety inhibits actual math ability and can present itself in students of all ability and achievement levels (Boaler, 2017; Maloney & Beilock, 2012). While some stress in school is a good thing, math anxiety comes from a negative stress, or “distress” (Martinez & Martinez, 2003).

Math anxiety can begin to develop at quite a young age, as early as the first grade (Ramirez et al., 2013). However, the transition from the elementary to middle grades appears to be a critical juncture in terms of the development of math anxious students. Field et al. (2019) found young adolescents who experienced lower mathematics achievement and increased anxiety during the middle grades were more likely to have lasting math anxiety through the rest of their schooling. Their perceived mathematical skill is closely related to their “status” in the classroom, which is built through social, peer, and academic standing (Wood et al., 2019). Between fourth and eighth grade, a student’s mathematical achievement, enjoyment of mathematics, and belief that “anyone can do mathematics” drops substantially (Martinez & Martinez, 2003). Similarly, Ma and Xu (2004) found that the middle grades are a critical period for the development of math anxiety in middle school boys. They found that math anxiety seems to develop across Grades 7 and 8, and after it has formed, it remains consistent throughout high school.

The root causes of math anxiety can be classified into three different types of variables: environmental, intellectual, and personality (Hadfield & McNeil, 1994). Environmental variables include a teacher’s instructional practices, a teacher’s own level of anxiety related to mathematics, and parental attitudes and beliefs about their child’s math ability (Eden et al., 2013). Intellectual variables include general cognitive ability and spatial reasoning. Individuals with lower spatial skills tend to have higher math anxiety (Ferguson et al., 2015). Personality variables with a relationship to math anxiety include self-efficacy (Akin & Kurbanoglu, 2011; Cooper & Robinson, 1991) and attitudes toward mathematics (Akin & Kurbanoglu, 2011).

Math anxiety can have an enormous impact on students. While there is debate about whether low math performance leads to high math anxiety or vice versa (or even whether there is a reciprocal relationship between the two), Ma (1999) conducted a meta-analysis of 26 studies looking at math anxiety and performance in mathematics and found a consistent, negative relationship. Ma and Xu (2004) found that this is particularly true for boys. This relationship is likely due to the finding that math anxiety impacts working memory, a critical component of mathematical cognition (Ashcraft & Krause, 2007). Students with math anxiety have more negative attitudes about math (Núñez-Peña et al., 2013), and a negative affect can inhibit creative problem solving (Isen et al., 1987).

Researchers have proposed a wide range of strategies to support students with math anxiety. While some have focused on strategies students can use to lessen their own anxiety (e.g., Jones et al., 2019), others have investigated classroom practices teachers can adopt to reduce student anxiety or prevent it from developing in the first place. Developing a learning environment focused on mastery rather than performance can reduce math anxiety in middle level students (Skaalvik, 2018). In classrooms structured to support mastery, students are intrinsically motivated to improve their own abilities, while in classrooms structured to focus on performance, students are motivated to outperform their peers and/or avoid mistakes (Furner & Gonzalez-Dehass, 2011). Teachers can encourage this kind of environment by allowing students to learn and perform at their own pace, which reduces the pressure to perform (Posner, 2011). Furner and Gonzalez-Dehass (2011) described how changing assessment practices to showcase growth over comparison and incorporating a variety of ways for students to demonstrate their knowledge can foster a positive learning environment that alleviates math anxiety. These practices all have a strong alignment with the basic principles of proficiency-based learning, suggesting that this kind of a system may be effective at reducing math anxiety.

Proficiency-Based Learning

Proficiency-based learning refers to a wide range of research-based practices supporting assessment and learning (Great Schools Partnership, 2016). While many of these practices have been used in some form for years, proficiency-based learning as a whole has risen in popularity over the last decade as educators across the country attempt to create more flexible and equitable assessment practices (Patrick & Sturgis, 2013). Vermont is one of 17 states that the International North American Council for Online Learning rated as advanced in terms of its work toward competency-based education (Truong, 2019). In 2013, Vermont passed the Flexible Pathways Initiative, which has motivated a major push toward proficiency-based learning in schools (Vermont Agency of Education, 2018).

At its core, proficiency-based learning is a paradigm that connects student performance to specific learning objectives. To guide schools in their implementation of proficiency-based learning, the Great Schools Partnership (2016) outlined a set of principles necessary to implement such a system effectively:

• Instruction should be centered on a set of clearly defined learning objectives;

• All assessments should be aligned with those objectives;

• Student progress should be measured relative to those objectives rather than in comparison to other students;

• Assessments should guide instruction;

• Proficiency on content is reported separately from habits of work;

• Grades are about communicating progress rather than rewarding student achievement;

• Students have multiple opportunities to demonstrate their learning without penalty;

• Students can demonstrate their learning through a variety of methods; and

• Students have a voice and choice in their learning.

A distinguishing feature of proficiency-based learning is the assessment-feedback loop. Traditional assessment systems tend to be unidirectional with teachers guiding instruction, assessing students on their learning, and compensating that learning with a grade. In contrast, proficiency-based systems are underscored with a strong belief that “student learning is enhanced by clear cycles of practice, feedback, assessment, and reflection” (Great Schools Partnership, 2018, p. 2). After learning objectives are clearly articulated, students engage in experiences designed to support their learning, they receive targeted feedback on their progress through formative assessments, and students are encouraged to self-assess and reflect on their learning. Then, if a student has not yet reached proficiency, the cycle repeats. Wormeli (2014) argued that this assessment-feedback loop is important for motivating young adolescents.

Because the specific practices teachers use when implementing proficiency-based learning can vary significantly, in this study we focused on five critical characteristics of the assessment-feedback loop: clear learning objectives, assessment for learning, feedback, reflection, and reassessment.

Clear Learning Objectives

An essential characteristic of the proficiency system is the development and implementation of clear learning objectives. These learning objectives not only guide teachers’ instruction, but they also serve as targets for students (Brookhart & Moss, 2014; Reed, 2012). When students have a clear vision of what they are expected to accomplish, they are better able to establish goals for themselves and make plans for how to achieve them (Moss & Brookhart, 2012). With clear learning objectives, middle level students are better able to monitor their progress and self-assess (Clarke et al., 2003; National Council of Teachers of Mathematics, 2014; Zimmerman, 2001). Having clear learning objectives shifts the focus of the classroom to learning or mastery rather than grades or performance (Black & Wiliam, 2010). Seidel et al. (2005) found that over the course of a school year, students whose high school physics teachers established clear learning objectives showed greater learning gains, described their learning environment as more supportive, and were more likely to experience self-determined motivation. Further, there is evidence in middle level mathematics specifically that involving students in the articulation of evaluation criteria can help establish a common expectation for learning goals and better support learning (Ross et al., 2002). Strong goal-setting skills, beginning with established learning goals, is an important contributor to both self-regulation and academic performance in middle level students (Falco, 2019; Zimmerman et al., 1992).

Assessment for Learning

One of the primary goals of traditional models of assessment is to evaluate and rank student achievement, and many students and teachers endorse this purpose (MacLellan, 2010). In such systems, teachers typically teach, then assess student learning, and then move on to new material doing little to nothing to support the students who did not master the previous material (Chappuis & Stiggins, 2002). This kind of system only allows those students who can learn at the expected pace to be successful, which leads to persistent inequities in the classroom. In response, many educators have reconsidered the structure of and intention for their assessments (Stiggins, 2005).

The primary focus of an “assessment for learning” model is to foster continual student learning and improvement by involving students more heavily in the assessment process (Chappuis & Stiggins, 2002). This way of viewing assessment has long been advocated by middle grades leaders (Stacki et al., 2020). Students participate in formulating criteria for evaluation, setting goals for their own learning, engaging in self-evaluation, and using feedback to determine areas for growth. The Association for Middle Level Education (AMLE) recognizes assessment for learning as a key assessment practice (Capraro et al., 2012). Classrooms in which teachers guide students through this process foster more self-directed student learning (Marshall & Drummand, 2006). When coupled with a focus on feedback rather than earning points, assessments for learning help to individualize instruction and promote learning (Doubet, 2012; Kaftan et al., 2006). Engaging in low-stakes assessment designed for students to practice new skills and get immediate feedback can even be enjoyable for middle level students (Nelson, 2016). Additionally, there is evidence that using assessment for learning in science and mathematics classrooms can improve student achievement (Wiliam et al., 2004).

Feedback

There is resounding evidence that, when provided effectively, feedback can have a significant impact on student learning (Hattie, 1992; Hattie & Timperley, 2007; Shute, 2008). Hattie (1992) synthesized 134 meta-analyses of educational interventions and found that “the most powerful single moderator that enhances achievement is feedback” (p. 9). Feedback can be delivered either orally or in writing (Strahan & Rogers, 2012). However, it is the content, not the quantity, of feedback that is most important (Black et al., 2004). Hattie and Timperley (2007) found that feedback is most effective when it helps students understand where they are in relation to learning goals and identifies processes to help them move forward. In a study comparing different types of feedback given to middle level mathematics students, Kramarski and Zeichner (2001) examined metacognitive feedback, which focuses on the problem structure and different problem-solving strategies, and result feedback, which provides guidance related to the final answer to the problem. They found that students receiving the metacognitive feedback learned more and produced better mathematical explanations than those who received results feedback. It is also essential that feedback be non-comparative, as young adolescents’ negative perceptions of mathematical capability are often an interpretation of feedback in the form of social comparison (Bandura, 1997; Falco, 2019).

Reflection

Reflection involves critical thinking about an experience and identifying possible changes for moving forward (Quinton & Smallbone, 2010). Young adolescents are developing their ability to reflect on their experiences and abilities, but these skills must be taught and integrated into their learning (Paris, 2016). Reflection is a vital component of mathematical cognition, and students are unlikely to reflect on their mathematical problem-solving processes unless prompted (Wheatley, 1992). Reflection supports the development of metacognitive skills through consolidation of new ideas and strategies (Dunlap, 1997). When a student reflects on their mathematical work, they are able to move outside of the action, giving themselves more control over their decisions (Wheatley, 1992). Structured reflections can also spark important student-teacher dialogue when a student poses a question or shares something that has confused them (Eisenbach, 2016).

Bond and Ellis (2013) compared middle level math students who learned using a reflective assessment to those who did not. In the five-minute activity, students were prompted at the end of a lesson to respond in writing to the prompt “I learned …, ” to share their ideas with a partner, and then to revise their written response. A nonreflective comparison group experienced identical lessons but with a five-minute review of the day’s work at the end instead of the reflective assessment. Students who completed the reflective assessment after each lesson had significantly higher achievement at the end of the four-week study than those that did not. Zimmerman et al. (2011) had similar results with at-risk college students learning and reflecting on mathematics.

Reassessment

It does not take much experience in the classroom to realize that all students do not learn at the same pace. Traditional assessment systems fail to take this into account and penalize students who develop conceptual understanding more slowly than their peers. O’Connor (2011) suggested that allowing students to reassess when they have not demonstrated proficiency and basing grades on the most recent evidence supports learning and growth. Though keeping track of resubmissions can be time consuming (Stange, 2018), teachers who use reassessment in their courses report that it helps all students meet challenging learning goals, particularly those with special needs (Knight & Cooper, 2019). Assessments which allow “ … yet” to be the norm (“I don’t get it … yet”) help to alleviate math anxiety in middle level students (Wood et al., 2019). Posner (2011) found that allowing students to resubmit work fostered a more positive attitude toward learning mathematics, and Esty and Teppo (1992) found that students stayed motivated in their math classes when they could reassess because they still had a chance to improve their grades. By allowing middle level students to learn at their own pace and to repeat assignments, the “all-or-nothing” mind-set which contributes to their math anxiety is alleviated (Martinez & Martinez, 2003).

Math Anxiety and Proficiency Based Learning

We hypothesized that proficiency-based learning could reduce the impact of students’ math anxiety for two main reasons: First, students who can reassess tend to report lower anxiety because of the lower the stakes for each individual assessment (Brackett & Reuning, 1999). Second, as described above, the key components of the assessment-feedback loop promote a mastery orientation, self-directed learning, and metacognitive activity. A learning environment driven by those ideas is likely to support students in continually working toward achieving learning goals, which also can alleviate math anxiety (Furner & Gonzalez-Dehass, 2011).

Methods

We set out to answer the research question: How do specific characteristics of proficiency-based learning interact with students’ math anxiety? We used a multiple case study approach. Because proficiency-based learning can look very different from classroom to classroom, we focused our study on two different middle level mathematics classes. We looked specifically at five students, three from one class and two from the other, who self-identified as being math anxious. We defined each individual student as separate cases.

School Contexts

This research was conducted in a very early stage of the transition to proficiency-based learning in two Vermont middle level mathematics classrooms in two different schools during the 2017–2018 school year. School 1 was a predominately White, mid-sized, suburban public school that served students in Grades PK–8, and 11% of students qualified for free and reduced lunch. In 2017, 72% of Grade 7 students in School 1 were considered proficient in mathematics, according to state-wide testing (Smarter Balanced Assessment by Grade Report, 2017). This is much higher than the state average of 44% proficiency in Grade 7 math (Vermont Agency of Education, 2017).

School 2 was also a predominately White, mid-sized, suburban public school that served students in Grades 6–8, and 17% of the school qualified for free and reduced lunch. In 2017, 49% of Grade 7 students were considered proficient in mathematics, according to state-wide testing (Smarter Balanced Assessment by Grade Report, 2017), which is much closer to the state average of 44% than School 1.

Selection and Description of Classroom Practices and Participants

The two teachers’ classrooms for this study were chosen because they were both relatively early adopters of proficiency-based learning. In a similar vein, both teachers worked to develop a classroom environment that focused on reasoning over correctness. Each teacher identified students in their classes who they perceived to demonstrate a high level of math anxiety, and each of the students in the study were also acutely aware of and self-identified as having math anxiety. Care was taken to select both male and female students and to select students who were articulate and willing to share.

This process yielded two teachers, Mary at School 1 and Karla at School 2 (all teacher and student names are pseudonyms). After we selected the teachers, five students were identified: Emily, Natalie, and Roger from Mary’s class and Patricia and Nathan from Karla’s class. All students in Mary’s class were 12 years old and in Grade 7, and all students in Karla’s class were 13 years old and in Grade 8. At the beginning of the year, both teachers described their implementations of proficiency-based learning and math anxiety. Table 1 gives demographic information and summarizes the in-place proficiency practices of each teacher in terms of each characteristic.

Table 1 Classroom Profiles

	School 1	School 2
Teacher	Mary	Karla
Age	52	60
Years Teaching	8	12
Subjects Taught	Math	Math and Social Studies
Middle-Level Education Endorsement?	Yes	Yes
Clear Learning Objectives	Objectives follow from projects. They are then clearly described in summative assessments.	For formatives, Karla will show what a 3 would look like, and what a 4 would look like, as well as provide a scale for each objective. These objectives come from common core and are rewritten into more understandable language. At the beginning of class, objectives for the day are projected on a smart board.
Assessments for Learning	For formative assessments, given weekly, students see what they understood and what they did not. A summative assessment is more an opportunity to give the teacher feedback.	Assessments are mostly formative, which are daily. Summatives are “low floor, high ceiling.” Skill baseline collected 3 times a year through standardized test.
Effective Feedback	Feedback addresses “what are the concepts they are not understanding?,” more than the individual mistakes.	Feedback tries not to “judge answers.” During class, teacher does not answer “is this right?.” Instead, she asks, “How would you know one way or another?” or prompts, “Work with their table.” The class stresses that mistakes help us learn.
Reflection	Reflections ask students “What do I know?” and “What do I need to know?” This reflection happens in class time when considering a project, and on paper during a summative assessment.	Students self-assess their work on formative assessments.
Reassessment	Students decide during the summative assessment if they need more work on particular topics. If so, they will try again during that period. If not, they will attach evidence of proficiency to their summative.	Students may revise work until they achieve a 3 (or a 4). If they don’t achieve a 3 on the first revision, they have a conversation with the teacher. They may then wait, and then retest in a different way. If the whole class does not understand, the teacher will reteach.

Data Collection

We collected data through a variety of means to be able to triangulate results (Creswell, 2021). These sources included semi-structured interviews, survey questions, observations, and artifacts from the classroom.

We conducted one-on-one, semi-structured interviews with each student that lasted approximately twenty minutes each, one near the beginning of the fall semester and a second at the end of the spring semester (see interview protocols in Appendix). The students were asked about each of the aspects of the assessment-feedback loop in the proficiency-based system and how they related to their math anxiety. Students also responded to a questionnaire with eight Likert-scale questions assessing their math anxiety, interest, and confidence in both the fall and the spring. In the fall, we also interviewed the teachers to gain an understanding of the specific practices used to implement proficiency-based learning. In the spring, we asked teachers to confirm, and revise if needed, the descriptions of their classroom practices to ensure accuracy and account for any changes that might have taken place over the course of the school year.

In addition to interviews, we conducted one mid-year classroom observation in which we took ethnographic field notes detailing the physical set-up of the classroom, the activities on the day of observation, and (most importantly) the interactions between teachers and students, particularly those students who participated in the study. This data set served to establish the context in which students experienced proficiency-based learning and math anxiety.

Finally, a combination of physical artifacts served as data for the study. Throughout the year, we collected formative and summative assessments—both copies completed by students and blank copies—as well as project rubrics and in-class handouts. These artifacts further illuminated how proficiency-based learning was executed in the classroom and how these ideas were communicated to students.

Data Analysis

Data were analyzed using qualitative coding techniques. For the first cycle of coding, we coded the interview transcripts with a set of a priori codes (i.e., clear learning objectives, assessment for learning, feedback, reflection, and reassessment) and also engaged in open coding in order cluster and analyze the data (Miles et al., 2014). We generated a data display of coded excerpts from each student’s interview related to each of the five characteristics of focus, and we used these data to begin drawing conclusions. Next, we analyzed the remaining data including field notes, classroom artifacts, Likert-scale surveys to triangulate our initial findings and look for emergent themes and sub-themes that would further establish the context (Miles et al., 2014).

Results

Clear Learning Objectives

Overall, the students found clearly stated learning objectives to be helpful and to alleviate their anxiety because they were able to determine what they needed to do to reach a certain level of proficiency. Learning objectives helped students identify and set goals for themselves, and having clear learning objectives helped students focus more on what they were supposed to learn and less on the grade they were going to get.

For example, one student discussed how clear learning objectives can help him set goals for himself, saying:

Well, it helps me, actually, a lot because it sets a goal for me. I’m like, ‘Okay, well, if I do this, then I’ll get a three, but if I do this, I’ll do get a four. But if I do this, I’ll get a two.’ And it sets a goal personally for me that, ‘Okay, I’m going to aim for three. And if I finish that in time, I’m going to try to aim for four. (See example of 1-4 grading scale in Figure 1.)

Figure 1. Example of Learning Objectives with 1 to 4 Scale.

In other words, the student was able to look at the clearly articulated levels of proficiency of the learning objective, select his personal target, and then take steps to reach that target.

In a similar vein, clear learning objectives allowed students to focus on learning. One student said, “They tell you in the learning target—instead of telling you how you did, they tell you what you need to work on.” For her, objectives and feedback focused her attention on what she got right on assessments and what concepts she did not yet understand, instead of just evaluating herself on her final numerical score.

While this clarity was overwhelmingly beneficial for most of the students, one student added that the objectives continued to fuel her anxiety stating, “I am definitely worried about them … Because I feel like I have to live up to that standard or else something bad will happen.”

Assessment for Learning

We found a distinct difference in the way students discussed the relationship between assessment and their anxiety by the function of the assessment. Assessments used for learning purposes, such as “skill checks” and small projects, decreased student anxiety and helped focus their attention on learning. As one student described,

[Assessments for learning] don’t really give me any anxiety because our teacher’s just like, ‘Okay, it’s okay. We’re just seeing where you are in math and how you’re feeling about it.’ And usually, it’s not really, really hard stuff to do, or we don’t have to do the hard stuff. We can just do what we feel comfortable doing.

Another student also found that smaller, more frequent assessments, such as practice worksheets, were helpful. She said:

Small assessments, we do those a lot, and they don’t really stress me out that much. I know that they matter because they can tell you where you are, I guess, in the unit that we’re doing … I know they’re important.

When assessments were more about providing the students with information than just capturing what they knew, the students were more involved in the process, and it was less anxiety provoking.

In contrast, students found assessment of learning to be stressful. A student described,

I don’t like the way that tests [cumulative assessments of learning] are put together because we don’t usually get to choose what we’re working on … you usually have to at least try everything because they want to see where you are at at specific points.

Another became overly focused on the content they struggled with, finding that summative assessments caused them to focus too much on what they could not do, causing them to do poorly on the things that they did know. In describing assessments of learning, they said,

I’m focused on that one particular thing instead of everything … I’ll be just focused on one thing that I need to learn. And I’ll get that right. And then the other things that I thought I knew, I won’t do as well.

Another student described how assessments of learning, in which she had to demonstrate knowledge on a final unit test, can be stressful saying, “I feel like there’s a place where I have to be at that certain time.” In general, students found that having a lack of choice and pressure to do well on things they know that they are not proficient in yet were their main sources of their anxiety.

Feedback

In general, students found clear, specific, and non-judgmental feedback helped to reduce their math anxiety. As noted earlier, effective feedback was incredibly important to the success of reassessment in proficiency-based learning. Specific feedback allowed students to develop a deeper understanding of what was expected from them. Students noticed that when they redid an assignment with a better idea of what to do to reach the learning targets, their anxiety was significantly lessened. However, some students reported that if they were asked to redo their work with no additional guidance, they could become frustrated and thus more anxious.

Specific feedback helped alleviate student anxiety. One student described how feedback on assignments can help them feel less anxious about math when “it’s really specific, and I actually understand it.” They explained, “I don’t know about other people, but I understand the feedback, but it’s really helpful to get that feedback instead of just saying, ‘You did this very well.’” Another student agreed, saying

Sometimes, [the teacher] just kind of writes, ‘Does this make sense?’ on the side of it, which isn’t exactly very helpful. It’s not a detailed thing that I can go off of to figure it out. Because it makes sense to me, but then she’ll make you second guess it.

So, receiving unclear feedback, particularly during a second or third attempt, had similar detrimental effects to math anxiety as no feedback.

To be helpful to students with math anxiety, feedback needed to be specific and also point to aspects of the work that students needed to improve upon. This was particularly important in a proficiency-based learning paradigm in which students were often reassessed. Students found that when reassessment and effective, specific feedback were used together, their anxiety was lessened. One student said, “I think it helps to have feedback on the paper so that I know where to start when I’m revising, if I revise.” One student expanded on the idea that feedback must point to what specifically needs to be improved. She described, “Well, definitely getting good feedback from my teacher really helps me … It makes me feel good because I know what I need to do and how I can do that.”

Additionally, students described how the context in which feedback is given can determine whether the feedback helps alleviate their math anxiety or increases it. Two students described their teachers pulling students aside during individual work time. While they liked the feedback and tutoring, one student said, “It still makes me really stressed because I feel like I’m not going to do well.” Another student agreed that the feedback itself was helpful, but he would feel nervous when the teacher would call him over, making him feel “singled out.” He reported that a lot of his peers felt the same way.

When feedback is clear, specific, and given in a safe context, it was among the most beneficial components of proficiency-based learning for math anxious students. For example, one student said that “definitely getting good feedback from my teacher really helps me … It makes me feel good …: This kind of feedback alleviated math anxiety.

Reflection

While students found reflection to be helpful for alleviating their math anxiety, and none of the students found reflection to increase their anxiety, it was not often cited to be particularly impactful. One student who found reflection to be helpful described, “I guess it usually makes me more aware of what I need to focus on and what I can do to improve that skill.” One student described how reflecting on their performance on their assignments while preparing for a second attempt was helpful: “And you can reflect on that test that we did, and what you did wrong, and fix it. So I think it’s pretty good.”

However, most students didn’t see a strong connection between their time spent reflecting and their anxiety about math. For example, in considering the impact of structured reflection, where students were asked to explain their thought processes in solving a problem and their perceived understanding of each part of the problem, one student described:

It doesn’t really [affect me]. I don’t know why we really have to do reflections because it, yeah, really just doesn’t help me. It just makes me think more deeply into it. But then I’m just like, ‘Okay. What is this? Why do I have to do this?’ So it doesn’t really help me out a lot. Yeah.

This student said that he understands why he was reflecting and took the opportunity to consider the problem more deeply, but he had a hard time seeing a direct impact on his learning. In general, students did benefit from reflection, but were not as enthusiastic and descriptive about the topic as compared to some of the other characteristics.

Reassessment

Students consistently cited reassessment to be the most, or one of the most, helpful characteristics for relieving their math anxiety. One student said,

Probably, for me, probably being able to redo my work [is the most beneficial]. It makes me feel pretty good when I can go through a test and redo it, and see what I did wrong, and not have to hand it in knowing that I got some questions wrong and not being able to redo it.

Another student agreed that being able to redo an assessment was the most beneficial to alleviating their anxiety, saying,

The one that’s probably the most helpful is being able to redo my work because it kind of gives me reassurance that I have another chance to fix it, and this time I can really ask for help, and I can fix all the things that I wasn’t doing so well on the first try.

Students who struggled with reassessment described how it could be helpful, as it gave them “a chance to show better work,” but could also make them feel that they are not doing well enough in general. These students described that effective feedback was essential for reassessment. One student described, “It makes me nervous when I don’t understand it at all, and when they say, ‘Redo this,’ but it doesn’t make sense to me, so I don’t know how to redo it.”

Multiple Characteristics Working Together

Students repeatedly reported how multiple characteristics of proficiency-based learning worked together to help them feel less anxious about mathematics. For instance, one student discussed the importance of effective feedback with reflection and reassessment, saying,

That would kind of fit in with getting feedback from the teacher and being able to reflect on the learning of my work. Because when you’re redoing your work, she normally goes over it with you sometimes. … And you can reflect on that test that we did, and what you did wrong, and fix it.

Other students agreed with this sentiment, describing how their math anxiety is lessened when teachers use specific feedback to help students reflect on their work, and then reassess to move toward clearly stated learning objectives. Another student agreed and stated,

I can look at the standards, and then look at my work, and see how I did, and kind of put a grade on it, and then see what my teacher thinks. So I think that’s pretty cool that I can do that.

He found that by having a good understanding of clear learning objectives, reflecting on his work, and receiving feedback helped him to do better when reassessed and helped to alleviate some of his anxiety about mathematics.

Table 2 summarizes results and shows which characteristics were particularly anxiety-alleviating and elements which may be anxiety-inducing.

Table 2 Summary of Results

Characteristic	Results
Clear Learning Objectives	Very anxiety-alleviating Helps students focus and set goals Shifts focus from a grade to learning
Assessment	Assessment for learning: Anxiety-alleviating Allows students to see “where they stand” in their learning Assessment of learning: Anxiety-inducing Students tend to focus on what they don’t know rather than what they do know
Effective Feedback	Clear and specific feedback: Very anxiety-alleviating Allows students to bridge gaps in understanding Vague feedback: Very anxiety-inducing
Reflection	Slightly anxiety-alleviating Reflection allows students to “step out of the action”, which can help to make the learning process less overwhelming to students with math anxiety.
Reassessment	The most anxiety-alleviating characteristic Particularly beneficial in conjunction with effective feedback and clear learning objectives

Discussion and Implications

The present study uncovered a relationship between each characteristic of proficiency based learning and middle level students’ math anxiety. We found that all characteristics (i.e., clear learning objectives, assessment for learning, reassessment, feedback, and reflection) were generally helpful for alleviating students’ anxiety about mathematics. Further, we found that each characteristic did not work in isolation but rather in conjunction with the others to support student learning. Each of the characteristics was made more powerful when it was combined with others.

A common theme in all results was the importance of a mathematics classroom culture centered on learning, rather than only achieving high grades, to help math anxious students. Each characteristic of proficiency-based learning showed the potential to shift the focus toward learning by allowing students to step out of the action and assess where they were in their understanding, where they would like to be, and how they planned to get there. This focus on learning seemed to help middle level students identify a meaningful purpose to their work in the mathematics classroom, which appeared to ease their anxiety.

Almost all students found clear learning objectives to be consistently helpful for alleviating their math anxiety. Students found that objectives helped them to focus and set goals. This finding was consistent with literature on goal setting. For example, Moss and Brookhart (2012) found that learning targets make lessons meaningful and help students to set goals and master challenges. Likewise, Seidel et al. (2005) found that learning goals help students feel supported in their learning and foster motivation to learn while also increasing their competence development over the course of the year. In middle level students specifically, Ross et al. (2002) showed that when these objectives are created as a class, students are better able to establish goals for their learning. The National Council of Teachers of Mathematics (NCTM) named “establish mathematics goals to focus learning” as one of eight key practices of effective mathematics teachers (National Council of Teachers of Mathematics, 2014). These goals “set the stage for everything else” (Hiebert & Grouws, 2007, p. 57). For example, Ross et al.’s students felt clear learning objectives allowed them to judge themselves on their learning rather than their grade (Ross et al., 2002).

The teachers in this study actively worked to build a classroom culture centered on learning over grading. Clear learning objectives and learning goals allow students both to self-assess and to be more focused during instruction (Clarke et al., 2003; National Council of Teachers of Mathematics, 2014; Zimmerman, 2001). When students are focused on what they need to learn, they may become more committed, effective learners (Black & Wiliam, 2010). To help students with math anxiety, teachers should develop and share clear, specific, and attainable learning objectives with their students.

Students in the present study found that assessment for learning alleviated math anxiety, while assessments of learning increased it, largely due to their cumulative nature. Assessments for learning are valuable checkpoints for middle level students, giving them an understanding of what they do and do not know (Capraro et al., 2012). One key strategy a teacher in this study used to ensure assessments would inform student learning was offering differentiated assessments and giving students the choice of which to complete. This is something that students cited as being helpful in alleviating their math anxiety. Assessment for learning is effective when teachers provide specific feedback and give students opportunities to reassess. Wormeli (2014) described how this combination of feedback and reassessment is critical as offering students detailed feedback without another opportunity to demonstrate learning can be frustrating for students.

On the other hand, the present study suggests that one of the most harmful parts of assessments of learning is their tendency toward a lack of choice and their comprehensive nature. Students tended to disproportionately focus their attention on problems they did not know how to solve, which hurt their performance across the entire assessment. For assessments to be beneficial to students’ math anxiety, students must feel that assessments are useful and supportive. Assessments of learning that do not allow students the opportunity to reassess also impose an artificial time constraint on student learning which has been shown to cause math anxiety (Boaler, 2017). Teachers can transform these types of assessments and use them for learning by using the assessment to identify gaps in understanding and give students opportunities to continue engaging with content while learning new material, and then teachers can offer the chance to reassess. In other words, teachers can work to make all assessments for learning.

Effective feedback was generally rated to be the second most anxiety alleviating characteristic of proficiency-based learning for students in the present study, behind reassessment. When done well, effective feedback may help students better understand their current abilities, and it illuminates strengths and areas for growth. Effective feedback also provides a way forward, helping students to begin to bridge the gap between their current understanding and their goal. It is possible that by providing a clear picture of a student’s current work and a direction forward, students feel more confident in their mathematics ability, and thus less anxious. Students in this case study shared that they preferred feedback that clearly told them what their mistake was and gave a hint or cue to get them on the right track. Hattie and Timperley (2007) identified this sort of clear and specific feedback as one of the top ten influences on student achievement, so it is not surprising that it is beneficial to math anxious students. It appears that feedback alleviates anxiety when it helps the student fill in their holes in understanding, and when students have the chance to fix their mistakes based on that feedback.

While students were less enthusiastic in their discussion of reflection than they were with other characteristics, any effects that reflection had were positive. No students reported detrimental effects of reflection on their math anxiety. This finding is consistent with literature, which suggests that reflection, via self-questioning, can be very helpful for students to the extent that they can monitor and improve their problem-solving skills (Calvin Gagnon & Maccin, 2007). One reason reflection may be helpful for students with math anxiety is because it gives them a chance to “step out of the action” and regroup before continuing with their learning (Wheatley, 1992). Black and Wiliam (2010) explained this idea, saying that before students can take action to improve their learning, they must reflect on where they stand and where they hope to be. However, students do not typically enter the middle grades with the ability to self-reflect. Helping middle level students develop the skills to evaluate their own progress toward their learning goals is critical to support the development of their understanding of mathematics (Bell & Pape, 2014). By extending the time for learning and allowing students the chance to regroup, reflection can be helpful to these students with math anxiety.

The most anxiety alleviating aspect of proficiency-based learning is reassessment without penalty. Reassessment extends the timeline of learning, allowing each student to learn at their own pace. This way, students can engage in initial learning and assessment without fear of failure. By moving away from averages and allowing for reassessment without penalty, students who take longer to learn are not penalized and remain motivated (Posner, 2011). Esty and Teppo (1992) argued that instead of using a system based on averages, which hurts students who have not yet been able to grasp the material early in the course, students should be given the time they need to learn and understand in order to decrease math anxiety. In order for math anxious students to successfully show their understanding, they must be given the time to reach that understanding with each topic. Posner (2011) found that students who re-submitted work in a proficiency-based system had higher scores and had substantial, positive change in attitude. Further, they began to take ownership of their learning and ask for assistance (Posner, 2011). Likewise, students in the current study found that reassessment gave them the opportunity to “really ask for help.” Students in this study generally found great comfort and satisfaction in being able to go back and fix their mistakes. Students then have the chance to do the best that they can without letting small issues phase them during an assessment.

Based on the findings from this study, we offer several suggestions for practices middle level mathematics teachers can use to decrease students’ math anxiety:

1. Establish clear, specific, and attainable learning objectives. Make sure that students know and understand these objectives.

2. Provide multiple levels of assessments for learning so that students can show where they are in their understanding. Stress to students that the purpose of these assessments is to see where they are in their learning.

3. Create assessments that allow students to show their understanding of particular topics without worrying about topics which they know they do not yet understand.

4. Allow students to redo their work without penalty in order to show understanding once it is attained.

5. Frequently give specific, clear, non-judgmental feedback.

6. Encourage students to reflect on their learning.

7. Give students the opportunity to reflect on and share their thinking. Use their explanation to identify misunderstandings.

The results of this study are widely applicable, as none of these practices requires fully adopting a proficiency-based approach. No matter the specific paradigm or grading scheme of a classroom, teachers can help to alleviate their students’ math anxiety by implementing the most beneficial aspects of proficiency-based learning.

Conclusion

The proficiency-based learning paradigm is helpful for alleviating math anxiety in middle level students. This is especially true for the characteristics of reassessment, effective feedback, and clear learning objectives. To best support students with math anxiety using a proficiency-based system, teachers should give students clear expectations and provide a variety of strong methods to achieve those expectations. Teachers should work with students to develop a stronger ability to determine what has been learned, and what needs to be learned. Once that is determined, students should have the opportunity to continue to learn, show their learning, and understand each topic presented in a mathematics class. Because mathematics is a subject that is constantly building on itself, students need to have a strong basis and proficiency in each topic along the way in order to feel like and be successful mathematics students throughout their schooling and beyond.

A major concern for many educators is that their math anxious student’s performance on state-wide standardized assessments. This study suggests that even with a strong proficiency-based paradigm, students are still anxious about assessment of learning. While the majority of an individual teacher’s assessment can be for learning, the standardized assessments of learning are outside of the teacher’s control. In a future study, it would be interesting to explore whether effective assessment for learning would eventually translate into improved performance on assessments of learning. Beyond performance, it would be interesting to see whether students eventually overcame their math anxiety to the point where even assessment of learning is not anxiety-inducing. Additionally, it would be interesting to reconsider the effects proficiency-based assessment with a closer focus on the relationship between math anxiety and testing anxiety.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Appendix

Interview Protocols

Teacher Interview

Instructions:

Please fully answer and elaborate, as you feel relevant, on each question. There is no correct answer, so speak freely. Feel free to ask questions if anything is confusing!

Background Questions:

• How old are you?

• How long have you been teaching?

• What subjects do you teach?

• Do you have a middle level education endorsement?

1. How do you implement PBL in your classroom?

2. Could you describe to me how you give feedback to your students?

   Follow-up:

   1. How often do you give feedback, and how complex is it?

3. Describe how you assess students (formative and summative) to check for understanding.

   Follow-up options:

   1. How often do you give formal tests/quizzes?

   2. What are tests like? Cumulative? Long? Can you do test corrections?

   3. How much homework do you assign, and how is it graded?

   4. Do you allow students to resubmit work? How many times?

   5. How are your students’ final grades determined?

4. How does math anxiety manifest itself in your classroom?

   Follow-up:

   1. Do you do anything to address it?

5. As is mentioned by the interviewee: Are you familiar with a growth v. fixed mind-set? Does it play a role in your teaching?

Fall Student Interview

Instructions:

Please fully answer and elaborate, as you feel relevant, on each question. There is no correct answer, so speak freely. Feel free to ask questions if anything is confusing!

Background Questions:

• How old are you?

• What grade are you in?

• Do you identify as male or female?

1. Do you like school? Why or why not?

2. What is your favorite subject? Why?

3. Can you describe to be how proficiency-based learning works in your math class?

   Follow-up:

   1. Have you ever had a math class that uses Proficiency-Based Learning before?

4. Do you think that Proficiency-Based learning systems help you to learn? How?

5. How does your teacher give you feedback?

6. Can you think of any specific things that your teacher does that makes you less anxious about math? a. More anxious?

7. Rate Agreement: I am certain I can learn everything taught in math.

8. Rate Agreement: I am confident I can do even the hardest work in my math class.

9. When I am in math, I usually feel (not at all at ease and relaxed, very much at ease and relaxed). Explain. Scale 1–7

10. When I am taking math tests, I usually feel (not at all nervous and uneasy, very nervous and uneasy). Scale 1–7

11. When the teacher is showing the class how to do a problem, how much do you worry that other students might understand the problem better than you?

12. Rate Agreement: Math will be useful for me later in life.

13. Rate Agreement: I enjoy doing math.

14. In general, how much do you worry about how well you are doing in math? (not at all, very much). Scale 1–7

15. Rate agreement: You can learn new things, but you can’t really change your basic intelligence.

16. As the interviewee mentions it: Are you a hard worker in math?

Note: Questions from this protocol were derived from the Middle Grades Longitudinal Study (2017).

Spring Student Interview

Instructions:

While I get set up, would you take a look at this for a minute or so? I am going to ask you about these things, so you might want to think about them for a moment. Feel free to jot down any notes.

Repeat Likert-scale questions from Fall Interview

General Question:

• How has your year in math class been?

Class-specific Questions:

Now I would like to ask you some questions about different aspects of your math classroom. For most of these questions, I will be asking how these aspects of the classroom affect how you feel about math.

Table

How does a big assessment (project, test, summative) impact your anxiety about math? How does a small assessment (“skill check”, written assignment) impact your anxiety about math? How does the way you are graded (1–4) affect your anxiety about math? Do you feel that you get options about how to show your teacher that you have learned material?

How does having the chance to redo your work (do another skill check during the summative) impact your anxiety about math? Does knowing that you will have the chance to redo your work (do another skill check during the summative) impact how you feel when you are doing an assignment for the first time?Karla’s students only: How do you feel about making mistakes in math class? Why? How does it affect your anxiety about math? Has your teacher ever met with you during intervention block when you don’t get a 3 on more than one shot? How does that affect your anxiety about math? If so, did you retest? Was it in the same way or a different way? Were you more or less nervous for this second test?

Does knowing the learning objectives for a class or assignment impact your anxiety about math? Do you ever read those? Do you think that they help you at all?Mary’s students only: Do you feel like you have a say in your learning objectives? How does that affect your anxiety about math?Karla’s students only: Do you find you can understand the learning objectives for this class? Does (understanding/not understanding) affect you anxiety about math? When your teacher shows you examples of what a “3” or a “4” looks like, does it affect your anxiety about the assignment?

When you get an assignment back, do you find the feedback written on it makes you feel better/worse? How else do you get feedback from your teacher? Does that make you feel better/worse?Karla’s students only: You might notice that your teacher doesn’t answer questions like “is this right?”. Does this affect your anxiety about math?

Do you ever have to reflect on your work and your learning for class? How? Does this impact your anxiety about math? Does grading yourself on assignments change your feels about math? Mary’s students only: You have to reflect before a summative. Does this affect your anxiety for the summative? For math?

Which, if any, of these helps you feel less anxious about math the most?

1. Being able to redo my work.

2. The way the tests and graded assignments are put together.

3. Being able to reflect on my learning and my work.

4. Getting good feedback from my teacher.

5. Having a good idea of the specific learning objectives.

Note: Questions from this protocol were derived from the Middle Grades Longitudinal Study (2017).