Mixed-ability teaching in an American High School led to higher attainment and improved relations between students, observed Jo Boaler

One of the most difficult challenges facing teachers of mathematics is the wide range of knowledge and motivation that different students have. Not surprisingly, many teachers support the practice of ability grouping so that they may narrow the range of and teach more effectively. In two different research studies I have conducted, in England and the US, I have followed students through secondary schools, investigating the impact of different teaching and grouping methods upon learning. In both studies the schools that used mixed-ability approaches resulted in higher overall attainment and more equitable outcomes. But in both cases the mathematics departments that brought about higher and more equitable attainment employed particular methods to make the mixed-ability teaching effective.

In this article I will describe the approach of Railside school, an inner-city high school in California. At Railside the students not only scored at high levels on tests, with differences in attainment between students of different cultural groups diminishing or disappearing while they were at the school, but the students learned to treat each other with respect. They learned to appreciate the contributions of students from different cultural groups, social classes, genders and attainment levels and develop extremely positive intellectual relations. I have termed this behaviour relational equity and this article will explain how it was achieved.

Our study of Railside school was conducted as part of a larger, four-year study of three US high schools. In the US students attend high school for four years, from when they are 13/14 to when they are 17/18. At Railside, the department employed a mixed-ability approach that is not used or well-known in the UK. The other two mathematics departments employed the equivalent of setting (called ‘tracking’ in the US) and traditional teaching methods. During the four-year study we collected a range of data, including approximately 600 hours of classroom observations, assessments given to the students each year, questionnaires and interviews. Railside school was more urban than the other two schools, with more English language learners and higher levels of cultural diversity (approximately 38% of students were Latino/a, 23% African American, 20% White, 16% Asian or Pacific Islanders; 3% were from other groups). On tests given to the students each year, the Railside students started at significantly lower levels than students at the other two schools but within two years they were achieving at significantly higher levels.

At the end of the second year (age 15/16) the average achievement of the students taught traditionally was equivalent to a GCSE grade D, while at Railside the average achievement was equivalent to a GCSE grade C. In other words, the Railside students learned in one year what it took the students taught traditionally to learn in more than two years. This result was achieved at Railside even though the students started the school at significantly lower levels, with severe gaps in their mathematical knowledge and understanding.

Students at Railside were also more positive about mathematics and took more courses. In the fourth year, 41% of seniors were enrolled in calculus (similar in standard to A-level mathematics), compared with approximately 27% in the other two schools. Importantly, inequities between students of different ethnic groups disappeared or were reduced in all cases at Railside whereas they remained at the other schools that employed tracking (for more detail see Boaler, 2008a, b, c).

Some mathematics departments employ group work with limited success, particularly because groups do not always function well, with some students doing more of the work than others, and some students being excluded or choosing to opt out. At Railside the teachers employed additional strategies to make group work successful. They adopted an approach called complex instruction, designed by Liz Cohen and Rachel Lotan (Cohen and Lotan, 1997) for use in all subject areas. The approach aims to counter social and academic status differences in classrooms, starting from the premise that status differences do not emerge because of particular students but because of group interactions. The approach includes a number of recommended practices that the mathematics department employed and refined for use in their subject area. In the next section I will review seven of the practices that the teachers employed and that our long term observations, interviews with students, and detailed analyses, showed to be important in the promotion of equity. The first four (multidimensional classrooms, student roles, assigning competence, and student responsibility) are recommended in the complex instruction approach, the last three (high expectations, effort over ability, and learning practices) were consonant with the approach and they were important to the high and equitable results that were achieved.

**Equitable teaching practices**

**Multidimensionality**

In many mathematics classrooms there is one practice that is valued above all others – that of executing procedures correctly and quickly. The narrowness by which success is judged means that some students rise to the top of classes, gaining good grades and teacher praise, while others sink to the bottom with most students knowing where they are in the hierarchy created. Such classrooms are uni-dimensional – the dimensions along which success is presented are singular. A central tenet of the complex instruction approach is what the authors refer to as multiple ability treatment. This approach is based upon the idea that expectations of success and failure can be modified by the provision of a more open set of task requirements that value many different abilities. Teachers should explain to students that no one student will be ‘good on all these abilities’ and that each student will be ‘good on at least one’ (Cohen and Lotan, 1977, p78).

At Railside the teachers created multidimensional classes by valuing many dimensions of mathematical work. This was achieved – in part – by giving students what the teachers referred to as group-worthy problems –– open-ended problems that illustrated important mathematical concepts, allowed for multiple representations, and had several possible solution paths (Horn, 2005). The teachers had created the algebra curriculum themselves, adapting problems from different curricula to make them group-worthy.

This enabled more students to contribute ideas and feel valued. When we interviewed the students and asked them ‘What does it take to be successful in mathematics class?’ they offered many different practices such as: asking good questions, rephrasing problems, explaining well, being logical, justifying work, considering answers, and using manipulatives. When we asked students in the traditional classes in the other two schools what they needed to do in order to be successful, 97% of them said the same thing – pay careful attention. This reflects a very passive act of learning. The different dimensions that students believed to be an important part of mathematical work at Railside were valued in the teachers’ interactions and the grading system.

The multidimensional nature of the classes at Railside was an extremely important part of the increased success of students. Put simply: because there were many more ways to be successful, many more students were successful. Students were aware of the different practices that were valued and they felt successful because they were able to excel at some of them. The following comments given by Jasmine, a student in the first year, in an interview gives an indication of the multidimensionality of classes.

*Student: ‘With math you have to interact with everybody and talk to them and answer their questions. You can’t be just like “oh here’s the book, look at the numbers and figure it out”.’Interviewer: ‘Why is that different for math?’*

*Student: ‘It’s not just one way to do it (…) It’s more interpretive. It’s not just one answer. There’s more than one way to get it. And then it’s like: “Why does it work?”’*

It is rare to hear students describe mathematics as more broad and more interpretive than other subjects. This breadth was important to the wide rates of success and participation achieved.

**Roles**When students were placed into groups they were also given a particular role to play, such as facilitator, team captain, recorder/reporter or resource manager (Cohen and Lotan, 1997). The premise behind this approach is that all students have important work to do in groups, without which the group cannot function. At Railside the teachers emphasised the different roles at frequent intervals, stopping, for example, at the start of class to remind facilitators to help people check answers or show their work or to ask the group questions. Students changed roles at the end of each unit of work. The teachers reinforced the status of the different roles and the important part they played in the mathematical work that was being undertaken. The roles contributed to the complex interconnected system that operated in each classroom, a system in which everyone had something important to do and all students learned to rely upon each other.

**Assigning competence**

An interesting and subtle approach recommended within the complex instruction literature is that of assigning competence. This is a practice that involves teachers raising the status of students that may be of a lower status in a group, by, for example, praising something they have said or done that has intellectual value, and bringing it to the group’s attention; asking a student to present an idea; or publicly praising a student’s work in a whole-class setting.

This practice was one that I could not fully imagine until I saw it enacted. My first awareness of it came about when a quiet Eastern European boy muttered something in a group that was dominated by two happy and excited Latina girls. The teacher who was visiting the table immediately picked up on it, saying ‘Good Ivan, that is important’. Later when the girls offered a response to one of the teacher’s questions he said, ‘Oh that is like Ivan’s idea, you’re building on that’. He raised the status of Ivan’s contribution, which would almost certainly have been lost without such an intervention. Ivan visibly straightened up and leaned forward as the teacher reminded the girls of his idea.

Cohen (1994) recommends that if student feedback is to address status issues, it must be public, intellectual, specific and relevant to the group task (Cohen, 1994, p132). The public dimension is important as other students learn about the broad dimensions that are valued; the intellectual dimension ensures that the feedback is an aspect of mathematical work, and the specific dimension means that students know exactly what the teacher is praising.

**Being responsible for each other’s learning**

A major part of the equitable results attained at Railside was the serious way in which teachers expected students to be responsible for each other’s learning. Many schools employ group work which, by its nature, brings with it an element of interdependence, but Railside teachers went beyond this to ensure that students took their responsibility to each other very seriously. One way in which teachers nurtured a feeling of responsibility was through the assessment system. For example, teachers occasionally graded the work of a group by rating the quality of the conversations groups had. In addition, the teachers occasionally gave group tests, which took several formats. In one version, students worked through a test together, but the teachers graded only one of the individual papers and that grade stood as the grade for all the students in the group. A third way in which responsibility was encouraged was through the practice of asking one student in a group to answer a follow-up question after a group had worked on something. If the student could not answer the question, the teacher would leave the group to further discussion before returning to ask the same student again. In the intervening time, it was the group’s responsibility to help the student learn the mathematics they needed to answer the question.

**High expectations**It was critical to the success of the students that teachers kept the demand of lessons intellectually high, both by providing complex problems and by following up with high-level questions. When students could not complete questions the teachers would leave groups to work through their understanding rather than providing them with small structured questions that led them to the correct answer.

In interviews with the students, it became clear that they appreciated the high demands placed upon them. The students’ appreciation was also demonstrated through questionnaires. For example, one of the questions started with the stem: ‘When I get stuck on a math problem, it is most helpful when my teacher…’ This was followed by answers such as ‘tells me the answer’ ‘leads me through the problem step by step’ and ‘helps me without giving away the answer’. Students could respond to each on a four-point scale (strongly agree, agree, disagree, strongly disagree). Almost half of the Railside students (47%) strongly agreed with the response: ‘Helps me without giving away the answer,’ compared with 27% of students in the ‘traditional’ classes at the other two schools.

**Effort over ability**In addition to the actions in which teachers engaged, the teachers also gave frequent and strong messages to students about the nature of high achievement in mathematics, continually emphasising that it was a product of hard work and not of innate ability. This message was heard by students and they communicated it to us in interviews, with absolute sincerity, as Sara (year 1) did:

*‘To be successful in math you really have to just like, put your mind to it and keep on trying – because math is all about trying. It’s kind of a hard subject because it involves many things. (…) but as long as you keep on trying and don’t give up then you know that you can do it.’*

In the year 3 questionnaires, we offered the statement ‘Anyone can be really good at math if they try’ and 84% of Railside students agreed with it, compared with 52% of students in the traditional classes.

**Relational equity**

It would be hard to spend years in the classrooms at Railside without noticing that the students were learning to treat each other in more respectful ways than is typically seen in schools and that ethnic cliques were less evident in the mathematics classrooms than they are in most schools. Further, such behaviour did not just happen to take place in a mathematics classroom; it was fundamentally related to the students’ conceptions of and work within mathematics. Thus, the work of students and teachers at Railside was equitable partly because they achieved more equitable outcomes on tests, with few achievement differences aligned with cultural differences, but also because they learned to act in more equitable ways in their classrooms. Students learned to appreciate the contributions of different students, from many different cultural groups and with many different characteristics and perspectives. It seemed to me that the students learned something extremely important, that would serve them and others well in their future interactions in society, which is not captured in conceptions of equity that deal only with test scores or treatment in schools. I propose that such behaviour is a form of equity, and I have termed it relational equity

**Conclusion**

Railside school is not a perfect place – the teachers would like to achieve more in terms of student achievement and the elimination of inequities, and they rarely feel satisfied with the achievements they have made to date, despite the vast amounts of time they spend planning and working. But research on inner-city schools, and the experiences of mathematics students in particular, tells us that the achievements at Railside are extremely unusual. In this artcile, I have attempted to convey the work of the teachers in bringing about the reduction in inequalities as well as general high achievement. In doing so, I hope also to have given a sense of the complexity of the relational and equitable system that they have in place. Teachers who have heard about the achievements of Railside’s mathematics department have asked for their curriculum so that they may use it, but while the curriculum plays a part in what is achieved at the school, it is only one part of a complex, interconnected system. At the heart of this system is the work of the teachers, and the many different equitable practices in which they engage.

**Further Reading**

- Boaler, J. (2008) ‘Promoting “Relational Equity” and High Mathematics Achievement through an Innovative Mixed Ability Approach.’
*British Educational Research Journal*, 34(2), 167-194. - Boaler, J, and Staples, M (2008c) ‘Creating Mathematical Futures through an Equitable Teaching Approach: The Case of Railside School.’
*Teachers’ College Record*, 110 (3), 608-645. - Cohen, E, and Lotan, R (eds) (1997) Working for Equity in Heterogeneous Classrooms: Sociological Theory in Practice.’ NY: Teachers College Press.
- Horn, IS (2005) ‘Learning On the Job: A Situated Account of Teacher Learning in High School Mathematics Departments.’
*Cognition & Instruction*, 23(2).