Nelson Mandela once said: “Education is the most powerful weapon which you can use to change the world.” (speech, Madison Park High School, Boston, 23 June 1990) And yet, millions of students across the globe are deprived of access to tools that could change their fate; chief among them: mathematical knowledge. The reasons behind students’ lack of success with maths can be attributed to the inequities that exist in their learning environments, particularly traits associated with their teachers, such as deficit-oriented thinking. This article aims to identify some of the most pressing concerns surrounding equity in mathematics education, including, a) tracking, b) flawed beliefs in teachers, and c) reform failures, and as such, attempts to offer some potential solutions to address these issues and suggest next steps.
Introduction
I suggest that one possible construction of equity involves understanding one’s opportunities to succeed, not just one’s access to resources. For example, access to the same amount of assistance for a student who is struggling with maths and one that is excelling, is not an equitable approach. In other words, by placing more emphasis and attention on the pupils who are not yet succeeding at rates aligned with their peers, I might be better able to understand the kinds of access to opportunities that may be beneficial. As such, systems and processes that hinder or infringe upon a student’s right to an equitable education and equal opportunities are inequities. Some of these inequities will be explored here in this article. My call is for future educators to endeavor to acknowledge them and work toward change.
Inequities
Strict Curriculums
Most teachers and schools follow a strict, standardized curriculum for mathematics. As such, mathematics is often taught in a linear fashion, too often using only one standard method for a specific type of problem. A myth surrounding this practice is its supposed efficiency, that this type of teaching leads students to “get it quick and get it right” (Dunleavy, 2018). Furthermore, many teachers perceive students who excel under this system as ‘being mathematically smart.’ Because the vast majority of students do not yet achieve at high rates in mathematics, it is not sufficient to simply believe in speed and accuracy as ultimately determining smartness. Not everyone is suited to the same, rigid curriculum. Thus, these rigidities are detrimental as they do not provide equal opportunities for all the students.
Rigid curricula do not offer students choices (such as multiple solution paths). In doing so, the rigid curricula can also foster a constrained relationship between students and mathematics, limiting students’ interactions with maths and affecting their performances in the subject negatively. Indeed, research has shown that students who learned under rigid systems scored lower on standardized tests, such as the SAT, and reported less socioeconomic opportunities than their counterparts (Boaler & Selling, 2017). One of the limitations of rigid learning is that it only allows students to solve problems already familiar to them, rather than different problems requiring similar approaches. This limited approach thereby limits the students’ intellectual freedom and suppresses their potential. On the other hand, those schools with non-traditional and more engaging curriculums, such as Railside School (Boaler & Staples, 2008), were more successful in providing equitable opportunities in mathematics education, as they offered more learning and interacting opportunities to students through a nuanced curriculum that attended to status and students’ identities .
Rigid curricula are also perceived as boring by students, thereby discouraging proper engagement with mathematics, let alone any interest in maths at all. With all these detrimental effects, rigid curriculum is a major obstacle for students to have equal opportunities to succeed in mathematics.
Tracking
Tracked systems are ones that offer mathematics at different levels for students with similar backgrounds. For example, Math Year 1 in the United States can be offered as an “honors” class or as a “regular” class, with “honors” being indicated for students who are perceived to be “more capable” or “smarter.” Alternatively, for maths in, say, South Africa, students are tracked into Sets, where you may be assigned to maths Sets, which are assigned based on merit. Students may move between sets based on their performance at the end of the academic year. Tracking causes major inequities in education. Because the most popular kind of tracking divides students into classes based on their perceived performances, students end up being tracked into “regular” instead of “honors” or “Set 2” instead of “Set 1” maths, not solely based on an indicator of their learning ability. Other systems group students based on their socio-economic backgrounds, or even on their race . Depending on those different tracks, students are taught different curricula, their progress is measured differently, and they have different access to resources. Usually, higher tracks get better treatment overall than lower tracks. It is clear that tracking allows resources to be distributed unequally, thereby limiting pupils’ chances and opportunities to succeed.
Tracking also entrenches the inequalities that already exist in the classroom, like students’ socio-economic backgrounds, as well as academic stereotypes based on race. Indeed, those assigned to lower tracks are often very likely to remain where they are (NCTM, 2014). Some students may lack self-confidence because of their lower-tracks; lowered confidence can, in turn, suppress a student’s underestimated abilities. The result is that the system can effectively destroy a students’ future social mobility, hindering their potential for progress towards a better future.
However, the solution to this problem should not just be detracking. Although detracking might first appear to move toward equality in a classroom and representation for all, it can be quite problematic as well. Detracking must be implemented thoughtfully for it to have a chance to be effective. Students might not integrate properly as a result of their different backgrounds, and this could result in poor cooperation and unnecessary tension between them. In these instances, equality is disregarded and academic priorities are lowered (Rubin, 2003).
Teacher Beliefs/Expectations
At many educational institutions, teachers often assign their pupils labels based on their perceived learning abilities and achievements. Horn (2007) found that teachers talk about their students as “fast,” “slow,” or “lazy” during department meetings. Although it may seem natural for teachers to talk about their perceptions of their pupils’ abilities, doing so has many inherent problems. Firstly, as mentioned above, these labels are false representations of those students’ actual abilities, especially as academic records do not necessarily equate to skills or understanding. For example, one student might be struggling simply because they are experiencing difficulties at home (e.g., financial crisis, emotional issues, etc.). Second, some students excel in maths simply because they are good at memorization. Regardless of the situation, the way teachers talk about their pupils directly affects their perception and their expectations of those pupils, leading to them subconsciously treating them according to a perception of their ability, and not their actual capacity for learning. Treating one according to perception is detrimental as it directly infringes upon the pupils’ rights to equitable treatment from their teacher. The practice of labeling students in this sense is no different to tracking, as students are unwillingly being grouped based on perception. It is therefore no surprise that such beliefs have the same detrimental consequences as tracking.
Moreover, teachers’ beliefs and expectations directly influence a student’s mentality in learning maths. Being viewed as a ‘slow’ learner has led pupils to see themselves as incapable of succeeding in maths and that maths can only be mastered by a few. This is incredibly harmful as these students then often match their teachers’ low expectations . The cycle then continues (NCTM, 2014).
Inequities are also present in how students are perceived as on or off task. Students who are perceived as misbehaving in class might instead be indicating their willingness to engage. By cracking down on those who are perceived as off-task students, teachers waste precious time and efforts they could be spending on engagement with maths (Hand, 2010).
Reform Failures
Although numerous research has been done on the aforementioned problems (citation needed if this claim stays), too few students still identify as being competent in and enjoying maths (Dunleavy, 2018). One main concern becomes how much those reforms contribute towards equity. Either reforms do not go far enough to address the issues (Martin, 2003; 2015), or they completely miss their supposed objectives. ‘Enlightened self-interest’ (Secada, 1989) is one of the most common of such failed reforms, in which reforms are introduced purely to combat social issues such as unemployment or an ailing economy, rather than a moral obligation to grant equal opportunities for all students. Such notions essentially render those reforms meaningless and lead reformers to cater only for the bare minimum extent to achieve their ‘practical’ goals.
Another harmful factor that furthers this reform failure is akin to the colonial idea of “White man’s burden”, in which ‘gains for minority groups coincided with White self-interests’ (Martin, 2015) to act as motives for reform. Such reforms from ‘White-benevolence’ simply cannot fit within the definition of equity, as their system is not set up for those minorities nor by those minorities. Therefore, any such reform is not legitimate on a principle level, despite any potential benefits.
Potential Solutions
Multiple Solution Paths
As discussed earlier, rigid curriculums that follow only one standard method effectively take away students’ choices in solving problems, which in turn creates an ineffective ‘one-size-fit-all’ system where few are set up to succeed. Many teachers only value students to get to the correct answer quickly through the ‘fastest’ method (Dunleavy, 2018), which is often not ‘fast’ at all for students who dislike it or barely understand it. In other words, efficiency of a method is relative. To mitigate this problem, teachers should value multiple solution paths, where all the methods to a particular problem are taught in detail. Here’s an illustration of a simple problem to demonstrate the effectiveness of multiple solution paths.
Q: 18 x 11 = ?
Answers:
18 x 11 = (20-2) x 11 = 20 x 11 - 2 x 11 = 220 - 22 = 198
18 x 11 = 18 x (10+1) = 18 x 10 + 18 x 1 = 180 + 18 = 198
18 x 11 = 1 (1+8) 8 =198 “ the 11 rule”
The above methods are all unique and different to the standard ‘ladder-method’(officially known as long multiplication), yet they all yield the same answer. Some would even argue the above solution paths are ‘faster’. By teaching all these methods, teachers empower students to have more mathematical tools when solving problems. Furthermore, it would prevent students from being trapped when forced to use the method that was ‘prescribed’ for them. They could choose any of the methods that suited them best. Additionally, this teaching style allows students to engage more with mathematics, not just in a linear fashion, but a multidimensional one too. They could even offer their own solution paths to the class, too; many teachers are actually surprised by what students could offer, which would enrich their expertise as well.
Multiple solution paths indeed give more opportunities to students, although it would be quite difficult to implement in humanities subjects. However, it fits perfectly within the mathematical context, as a variety of different approaches could be utilised to solve a single problem, especially in the fields of algebra and geometry.
Mandatory Support
Because so many inequalities already exist within a classroom, teachers should do more than just teach and should intensify their work to provide equal opportunities for all their students (Apple, 1992). This should come in the form of mandatory support for their pupils, whenever they need any form of help. A non-compulsory consolidation period could be set up after school for teachers to answer students’ questions. If such a system already exists, then it should be kept; however, in the absence of such a system, a new support system akin to the one mentioned above should be established.
Furthermore, we argue for teachers to have built in time to give more attention to the students’ lives outside the classroom. This is especially important for students of colour and students from marginalized backgrounds that may have outside issues which could contribute to their poor performance in class. By showing concern, empathy, and care teachers could develop better relationships with these students, which could lead them to engage more in class to counteract the outside stress that is negatively influencing them. When students and teachers get to know one another, they would be able to congratulate one another on their achievements outside academics, which would boost their confidence and sense of belonging within the class environment (Horn, 2017).
Lesson Meaningfulness
To make maths more meaningful, one has to deviate from its dull, abstract concepts and relate them to more practical means. This way students would not see maths as boring, but rather something that could be useful and something that they could relate to in their daily lives. This would give students equal opportunities to understand. We offer one example that may make a more abstract concept appear more authentic.
The field of calculus is one that students often find to be monotonous and they see little value in learning such abstract concepts. It is also one that they struggle to grasp with. Despite this, there's more to calculus than just derivatives and integrals. For instance, these concepts can be easily integrated into a simple physics framework. The basics of displacement, velocity, and acceleration can be clearly seen as derivatives/integrals.
Figure 1. Calculus Example
As illustrated in Figure 1, the graphs show that the gradient/derivative of displacement is velocity, and the derivative of velocity is acceleration. Similarly, the area under the curve shows the integral of acceleration is velocity, and the integral of velocity is displacement. By teaching calculus in this fashion, teachers effectively show the real-world applications of calculus, which would make the abstract concepts easier to understand and easier to relate. However, it must also be noted that some mathematical concepts are difficult to translate into everyday language and situations, as seen in some of the unnecessary arguments that developed during these more ‘authentic’ explanations (Godfrey & O’Connor, 1995).
Classroom Belongingness
Many students have a false belief that mathematics is an inherent skill that only the selected few could master. Indeed, this notion has been reinforced by their parents, who claimed that they were not good with maths either (Horn, 2017). The effect of this is that many students are hesitant about maths as a subject. To resolve this, teachers should make their learners’ classroom environments as accommodating as possible, in other words, making them feel belonged. Students should have a classroom ‘identity’, where they have specific ‘roles’ within a class (Nasir and Hand, 2008). This is important as nobody would feel ‘left out’ and more engagement could occur. Ultimately, this would lead to academic success for some of the students.
As already mentioned above, mandatory support, both academically and morally, would be crucial to make students feel more belonged. This should also include learning names, and identifying something common between students and teachers. (Horn, 2017). Furthermore, teachers should not punish students harshly on disciplinary issues, or waste time correcting students’ inconsequential mistakes like spelling. Lastly, maths should be made more entertaining and fun, as to counter some of its dullness. This could involve class discussions on different solution paths, interesting exploration activities (such as non-standard units in Godfrey & O’Connor, 1995), and even small projects synchronised with the curriculum. Of course, making abstract concepts more relatable and practical -- as mentioned above -- would also help students to ‘fit in’.
A very interesting study (Nasir & Hand, 2008) compares student behaviours in the classroom to their behaviour during basketball practices in an attempt to figure out what motivates student participation. The study points out that fostering intimate relationships between students and their teachers plays a vital role. Specific traits such as making mistakes acceptable, giving specific roles could enhance student participation and their learning experience. Another important factor is the process of teaching other novices (students helping out one another). Students will not feel ashamed to ask their peers, and this practice would reinforce the mathematical knowledge they shared, especially when the teacher is busy helping others (Nasir, 2008). The study also showed a feedback system could be borrowed from basketball practices, where the teacher could constantly give suggestions on how to improve.
I could testify to these benefits from my own experience as a student. My maths teacher from Grade 8 to Grade 9 really incorporated the above elements well into his classroom. While teaching, he often creates very interesting maths problems that involve his students and their different personalities. Sometimes he would even call upon those students to solve the problems involving themselves. This strategy was very effective, as everyone in the class felt they were important to a certain extent and that they have specific, albeit fictitious, roles. It also greatly entertained the class, seeing their fellow classmates being depicted in those example problems. Furthermore, he often rewarded us with his home-made brownies whenever the class performed well in a standardized test. All this enabled the whole class to feel belonged during maths periods, encouraging their participation and fostering an environment where everyone is eager to learn. Indeed, all my classmates felt included in a classroom community, showing that we have indeed been given equal opportunities to succeed.
Changing Beliefs
Though it is important to implement all of the above, it must be noted that the most crucial thing is for teachers to change their inherent beliefs. Afterall, without the right intentions, true equity will never be realised. Teachers must bear in mind that whatever they are doing is for the benefits of their students, for them to have equal opportunities to succeed. Any wrong intentions, such as enlightened self-interest (Secada, 1989), will result directly as failed reform, as discussed earlier in the article.
Teachers must change their false beliefs about their students, in order to treat them equally without any bias. Beliefs could be categorized into two categories: productive and unproductive (NCTM, 2014). Productive beliefs are ones that would help teachers to promote equity while unproductives ones hinder students’ opportunities to learn maths. For example, a productive belief is that all students, regardless of race, gender, class, religion, sexual orientation and other identity traits, are capable of succeeding in maths. An unproductive belief would be that only certain students, with innate intelligence and ability, could excel at maths. Other unproductive beliefs include tracking, categorising and labeling students, and equity as equal treatment only. It is clear that productive beliefs must be spread while unproductive ones should be criticized. In other words, the first step to equity is to change some of the false beliefs teachers might have, even if they are subconscious.
Conclusion
To ensure all students have equal opportunities to succeed, teachers should go beyond their duty of simply teaching within a classroom. They should make their classroom environments more accommodating for students via more meaningful and more interesting lessons, ensuring everyone would be engaged and motivated to learn. They also need to recognise which beliefs are productive and which are not, as well as problems that hinder the progress of equity such as tracking. Moreover, they should offer mandatory support to students, as well as giving multiple solution paths. These actions would certainly give students a better chance at success.
However, true equity can only be realised through a grassroot effort (Moses & Cobb, 2002), where teachers must cooperate to ensure students have all the support they need, and that equity practices are spread and implemented as much as possible. Perhaps they could even come up with a standardised equitable curriculum.
References
Apple, M. W. (1992). Do the standards go far enough? Power, policy, and practice in mathematics education. Journal for Research in Mathematics Education, 412-431.
Boaler, J., & Selling, S. K. (2017). Psychological Imprisonment or Intellectual Freedom? A Longitudinal Study of Contrasting School Mathematics Approaches and Their Impact On Adults' Lives. Journal for Research in Mathematics Education, 48(1), 78-105.
Boaler, J., & Staples, M. (2008). Creating mathematical futures through an equitable teaching approach: The case of Railside School. Teachers College Record, 110(3), 608-645.
Dunleavy, T. K. (2018). High school algebra students busting the myth about mathematical smartness: Counterstories to the dominant narrative “get it quick and get it right”. Education Sciences, 8(2), 58.
Godfrey, L., & O'Connor, M. C. (1995). The vertical hand span: Nonstandard units, expressions, and symbols in the classroom. The Journal of Mathematical Behavior, 14(3), 327-345.
Hand, V. M. (2010). The co-construction of opposition in a low-track mathematics classroom. American Educational Research Journal, 47(1), 97-132.
Horn, I. S. (2007). Fast kids, slow kids, lazy kids: Framing the mismatch problem in mathematics teachers' conversations. The Journal of the Learning Sciences, 16(1), 37-79.
Horn, I. S. (2017). Motivated: Designing math classrooms where students want to join in. Heinemann.
National Council of Teachers of Mathematics. (2014). Principles to Actions: Ensuring Mathematical Success for All, Author.
Martin, D. B. (2003). Hidden assumptions and unaddressed questions in mathematics for all rhetoric. The Mathematics Educator, 13(2).
Martin, D. B. (2015). The collective Black and principles to actions. Journal of Urban Mathematics Education, 8(1).
Moses, R., & Cobb, C. E. (2002). Radical equations: Civil rights from Mississippi to the Algebra Project. Beacon Press.
Nasir, N. I. S. (2008). Everyday pedagogy: Lessons from basketball, track, and dominoes. Phi Delta Kappan, 89(7), 529-532.
Nasir, N. I. S., & Hand, V. (2008). From the court to the classroom: Opportunities for engagement, learning, and identity in basketball and classroom mathematics. The Journal of the Learning Sciences, 17(2), 143-179.
Rubin, B. C. (2003). Unpacking detracking: When progressive pedagogy meets students’ social worlds. American educational research journal, 40(2), 539-573.
Secada, W. G. (1989). Agenda setting, enlightened self‐interest, and equity in mathematics education. Peabody Journal of Education, 66(2), 22-56.
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