# Further Reflections on the Questions

Having reviewed some of the major facets of multicultural education, I continue to work backward to address plausible answers to the other questions posed at the beginning of this chapter. I bravely present views as well as relevant research in the hope that they will serve as catalysts for readers’ thoughtful reflections and discussions.

*6. Are there cultural groups for which the traditional approach enhances students’ achievement?*

Stiff (2001) writes: “Of course, many students are able to make sense of the mathematics when it is organized in a traditional framework. The best of these students are seen as special and talented. Hence, the point of view that mathematics can be understood only by the chosen few is widely accepted” (3). Unfortunately, *best*, *special,* and *talented* generally have not been adjectives to describe the performance of minorities in mathematics, and a reason for that may be attributed to traditional teachers who pay little or no attention to students’ learning styles or cultures because there is very little variation, if any, in their direct teaching method. They assume that students will succeed in mathematics if students work hard enough to memorize the class lectures, do the homework problems, and correctly answer the test-type routine questions.

However, some students taught by teachers who consider themselves traditional do well on both the basics and higher-level thinking skills. A classic example is Jaime Escalante, the legendary Los Angeles teacher who was highly successful in getting a large percentage of low-achieving minority students to pass the Advanced Placement calculus exams largely using traditional strategies. However, in the movie *Stand and Deliver* (Warner Brothers, 1988), Escalante did employ nontraditional (i.e., Standards-based) tactics to motivate his students. In one scene, he used his fingers as a manipulative for multiplying by nine—an action that quickly captured the interest of one of the students. In another, he demonstrated his belief that infusing culture in his curriculum would motivate students to learn by telling his Latino students that they had a rich heritage: They were the descendant of the great Mayans who were the first to use zero in mathematics, something neither the Greeks nor the Romans had devised. He further tells his students that they were “heirs to a great mathematical tradition” and that they could do mathematics because, he said passionately, “It’s in your blood!” His strategies and dedication served to show the world that students could succeed with a teacher who truly believes they are brilliant while simultaneously providing them with opportunities to excel. In 1999, he was inducted into the National Teachers Hall for Fame for his success in getting students to believe in themselves, and in 2016 the U.S. Postal Service issued a Forever Stamp in his honor (http://wwwbiography.com/people/jaime-escalante-189368#synopsis). Given that some traditional methods do work, the next question is: why not for all students? Can this method help improve the scores of students who underperform? The use of traditional methods to teach students assumes at least three perspectives: first, that all students learn from the information given to them; second, that life experiences, race, socioeconomic status (SES), and learning style have no bearings on students’ processing of information; and third, that, since no alternative teaching approach is offered to failing students, it is then the students’ fault that they are failing or that those who cannot learn in this way are just dumb, lazy, or lacking a math brain. Educators who do not blame students for failure have conducted research to find out why so many fail (Boaler, 2002; Gutierrez, 2002). Their research shows results that are at the heart of the NCTM’s recommendations: Teachers must be well prepared to teach, and they must teach in ways that recognize and appreciate the fact that students come to the classroom with different cognitive ways of processing information and that their different backgrounds influence how they come to know and understand what is taught. Thus, to make traditional teaching work for all students, teachers need to ask, “How prepared are the students to engage in traditional teaching?” Students may need to be taught the skills for succeeding in that environment, which are to listen to the teachers’ information, copy notes from the board, answer questions correctly and ask questions, follow directions, do assignments, and study to pass the tests—but students have been practicing these skills in traditional classes since first grade! I contend that more of the same, again, is not likely to make any difference.

Because the traditional way may not work for some, it follows that another option is to change the teaching method rather than give students more of the same. Observations of how the students work and process information provide data for teachers to determine students’ preferred learning styles. By varying their teaching strategies, teachers begin to move outside of traditional approaches and into the SBS domain. Doing so transports the teachers’ methods out of the traditional set in Figure 4.1 and into the SBS area. When teachers believe that all students can learn, they focus on *how* their students learn, assess what their needs are, and then adapt their teaching methods to best facilitate the learning.

*5. Will students’ achievement be enhanced if a teacher uses a mixture of SBS and traditional strategies as in Set B of Figure 4.1?*

NCTM recommends that teachers use a variety of approaches, but is the traditional method one of those approaches? Definitely. Past NCTM president Lee Stiff (2001) writes, “The Standards documents recognize that successful mathematics education programs include the best of traditional and reform-based mathematics education” (3). Teachers using strategies in set B of Figure 4.1 may be knowledgeable about the standards and traditional teaching and are choosing the two based on the needs of their students. If so, set B is the same as set A: Both are Standards-based. This question thus becomes the same as question 2 and is answered later. If not, then set B teachers may be in transition from using one method to the other and need more guidance.

*4. Are there subsets of SBS that work best for enhancing the achievement of some groups as in set A of Figure 4.1?*

There are subsets of SBS that research reports work well for some groups, as readers will see in the profiles of this book. However, strategies for teaching students from different cultures cannot be neatly boxed under the label: “Look *here* for strategies that will always work for particular groups.” Making generalizations about any group falsely assumes homogeneity of life experiences among members. Using Latinos as an example, Moschkovich (1999) writes:

Latino students come from diverse cultural groups and have varied experiences. It is difficult to make recommendations about the needs of Latino students in mathematics that would accurately reflect the experiences of students from a remote Andean village, a student from a bustling Latin American city, a student from a Southwest border town in the United States, and a student from an Afro-Caribbean island. Although these students will have some shared experiences such as some relationship to the use of Spanish, there will also be many differences among these students’ experiences, either at home or at school. Both the differences and commonalities among Latino students should be kept in mind when designing mathematics instruction.

*(10)*

There are, however, suggestions from research on strategies that have benefited the learning of specific groups of students that are demonstrated in this book. Teachers can start there with similar groups and modify where necessary.

*3. Will students’ achievement be enhanced if a teacher teaches from a traditional perspective as in set T?*

This question is already addressed in question 6 where the discussion there suggests that the answer is no.

*2. Will students’ achievement be enhanced if a teacher teaches from a Standards-based (SB) perspective?*

Most of the research supports a yes answer to this question. In response to the NCTM’s *Standards* (1989) call for a shift from traditional teaching to SB methods, the National Science Foundation (NSF) in the early 1990s funded the development of curriculum materials to facilitate school adoption of Standards-based curricula and processes. The research on these NSF-funded curricula projects shows that, in general, students perform at comparable or greater achievement levels than the comparison group taught with traditional curriculum. For research results on elementary programs, see Fuson et al. (2000), Riordan and Noyce (2001), Budak (2015). For research results on the middle school curriculum, see Riordan and Noyce (2001), Reys et al. (2003), Tarr et al. (2008). For research results in high schools, see Grouws et al. (2013), Tarr et al. (2013), Saddler (2015). While all of these studies show the important contribution of curriculum to improving students’ learning, Tarr et al. (2008) also shows the important contribution of teachers using SBS: The NSF-funded curricula students in his study performed as well as the comparison group, but they *outperformed* the comparison group with teachers implementing that curricula *together with* SBS strategies. They conclude that what is needed is *“coherence *between the textbook and implemented curricula; that is, consistency between curriculum and instruction is needed in order to actualize student learning in mathematics” (275). This research suggests that SBS are effective for helping students to achieve. But do they do so *equitably*? I will use the definition of *equity* from a report by Noguera et al. (2015) that defines equity as “the policies and practices that ensure that every student has access to an education focused on meaningful learning, taught by competent and caring educators who are able to attend to the student’s social and academic needs, and supported by adequate resources that provide the materials and conditions for effective learning” (3).

Research on whether SBS can promote equity by reducing the gap between lower and higher mathematics performance of SES students has not been promising. McCormick’s (2005) dissertation study showed that, while low SES and minority student scores showed *insignificant* gains when using SB curricula, this was not true for the higher-SES white students. Grady’s et al. (2012) study indicated there was no significant difference between the SB and traditional curriculum across all areas that included low SES and minority students, with one exception—the algebra subtest results—in which case, the SB students performed lower than their peers who were taught with a traditional approach supplemented with a computer software called Mountain Math (44). Budak’s (2015) study shows that students taught with NSF-funded curricula perform as well as or better than those taught with traditional curricula. However, within those taught with the NSF-funded curricula, middle- to high-SES white students had significantly higher scores on the assessment than the low-SES, African American, Latino/a students.

I next report on three classic studies that questioned whether SBS is appropriate for students from low socioeconomic status and minority students. One study considers implementing SBS in reading, an area undergoing reform similar to mathematics, and the other two are in mathematics. In recounting her experiences in trying to teach her class of white and black students through SBS for reading (in this particular case, an open-ended classroom situation using learning centers), Del- pit (1995) expresses her exasperation at the level of achievement of her black students.

My white students zoomed ahead. They worked hard at the learning stations. They did amazing things with books and writing. My black students played the games; they learned how to weave; and they threw the books around the learning stations. They practiced karate moves on the new carpets. Some of them even learned how to read but none of them as quickly as the white kids. I was doing the same thing for all my kids—what was the problem?

*(13)*

Because she cared and believed her students could learn, Delpit then searched for answers and eventually read research reporting that because the norms of SB classrooms are consonant with white middle-class homes, students from those home come to school better prepared to learn from SB strategies. For example, indirect or open methods of communications are common in such families, whereas black students from low-SES or working-class families are accustomed to direct communications and explicit facts. Delpit concluded that her black students did not have the necessary basic skills to benefit from the open-ended situations of SBS as she presented them. They needed more direct practice on writing skills. However, she firmly rejects the notion that a traditional drill- and-kill approach to instruction is best for minority students. Indeed, she states that “skills are a necessary but insufficient aspect of black and minority students’ education” (19). In her interview with Goldstein (2012), Delpit reaffirms: “One cannot divorce the teaching of basic skills from the demands of critical thinking; having kids question what is in newspaper articles, even question what is in textbooks” (1). Of black children, Delpit (2012) goes further to say, “If we do not recognize the brilliance before us, we cannot help but carry on the stereotypic societal views that these children are somehow damaged goods and that they cannot be expected to succeed” (5). According to Danny Bernard Martin, her use of the word *brilliant* to portray black children served as a motivation to include the word *brilliance* in the book he coedited with Jacqueline Leonard (2013) (https:// wwwyoutube.com/watch?v=tYneVA8kBp0).

Lubienski (2000) conducted an action research project with 30 students to examine the problem-solving reaction of white seventh-graders of different SES to the learning of mathematics through open problems presented within a context and taught using SBS. She found that, while higher-SES students had more confidence and a sense of where to go with the problem, lower-SES students wanted more external direction and often missed important points in the problems. In addition, she found that higher-SES were more interested and persevered when doing problems, whereas lower-SES found the mathematics less interesting than the activities involving games or contexts of interest to them (477). Lubienski cautions readers not to generalize or conclude that her study of only 30 students implies that lower-SES students will learn less from SBS strategies. One of her conclusions is that scholars need to be aware that, while SBS could improve both lower-SES and higher-SES students’ understanding of mathematics, they could also increase the gap in their mathematics performance. She cites the work of Hess and Shipman (1965), who report differences between the way middle- and working-class mothers helped their children solve problems: While working-class mothers used a traditional approach of merely telling them how to do it, middle- class mothers, using an approach similar to the philosophy of SBS, asked questions to help their children determine the key features of the problem (475). Thus, while she sees the algorithmic traditional teaching as dull and ineffective in promoting conceptual understanding, Lubienski (2000) suggests that it may be better for promoting equity because it provides “a relatively level playing field by having clear rules and being equally disconnected from all students’ realities” (478).

Both Delpit’s and Lubienski’s work report incongruence between the norms of low-SES students and those valued in classrooms using SBS. Delpit rejects a return to the traditional approach, and, although Lubienski explicitly suggests that option, she is careful to present other research reporting success in that area before stating that “methods that are promising for many students could pose unexpected difficulties for students who most need mathematics empowerment” (480). Seven years later, Lubienski (2007) reflects on her research and adds:

Although this research is sometimes criticized for promoting simplistic stereotypes, avoiding discussions of class differences is detrimental for low-SES students, whose strengths and needs we might then ignore. I am not suggesting that the distinctions above are true for every family or that children should necessarily receive instruction that matches their home environment. Indeed, one could argue that low-SES students are *most* in need of mathematics instruction that emphasizes questioning and problem solving. The point here is that mathematics teachers should pay attention to the particular orientation toward learning with which children have been raised, particularly when trying to implement instructional reforms.

*(1)*

Can we promote equity for low-SES students by using SBS? Just as for traditional teaching, teachers using SBS need to ask the analogous question, “How prepared are the students to engage in SBS?” Cases to consider for answering this question are as follows:

- 1. If the basic content skills are lacking, then the option recommended by Delpit and NCTM is that they be taught within the context of critical and creative thinking. This idea can be extended to a recommendation that teachers teach students whatever skills they lack as they
*keep moving forward*with learning. For example, middle-grade students lacking basic mathematics skills can revisit and practice those skills in the process of solving challenging problems requiring, say, algebraic equations. What allows this strategy to work for those students is the teacher’s intention to also make basic skills a focus in those problems. - 2. Since Lubienski (2000) mentioned that low-SES students’ interests were high on problems of interest to them, then a second option is to find or create such problems that simultaneously develop critical thinking.
- 3. If the skills students need to succeed include
*how to work*in an SBS environment, then a third option is to teach students how to participate in that environment. In addition to the traditional learning practices listed in question 6 for traditional teaching, there are new ones for students to learn that include the ability to explain and justify answer(s), to teach to and learn from cooperative groups, to know when and how to take notes during discussions, to make conjectures, to seek, gather, and analyze their own data for testing their conjectures, and to apply a variety of strategies to open-ended situations.

4. A fourth option is to use Lubienski’s suggestion of having low-SES students study under the traditional approach since it reduces the complexity by giving them explicit instructions to well- defined problems. But in this case, however, teachers will have to find other means to teach students critical thinking skills and this is problematic since the SMP should support problem solving.

The first three options for helping students succeed within an SB classroom environment remain within the domain of SBS in Figure 4.1. The fourth option raises concerns for Boaler (2002), who in her own research on the relationship between equity and reform curriculum, discusses Delpit and Lubienski’s work. She writes:

The idea that some students may be disadvantaged by some of the reform-oriented curriculum and teaching approaches that are used in schools is extremely important to consider and may reflect a certain naivete in our assumptions that open teaching methods would be accessible to all. But whilst the realization that some students may be less prepared to engage in the different roles that are required by open curriculum is very important, analyses that go from this idea to the claim that traditional curriculum are more suitable may be extremely misleading. This is partly because they reduce the complexity of teaching and learning to a question of curriculum, leaving the teaching of the curriculum relatively unexamined.

*(2)*

Thus, Boaler sets out to research how such students can be helped to work in an SBS environment for which they are not prepared. In 2002, she reviewed her own previous research supporting equity (Boaler 1997) and then questioned why other SBS research focused on equity, such as Lubienski’s, yielded contradictory results. Using Gutierrez’s (2002) research suggestion that teachers’ practices may be a key for establishing equitable SB learning environments, Boaler (2002) designed a study for white, low-SES students’ achievement on open-ended problems. In England, she studied students from two high schools in low-income areas where underachievers and overachievers were equally divided across SES. Teachers from one school of about 100 of her subjects were knowledgeable about and had been applying SBS for two years. The other school of about 200 subjects used the traditional method. Rather than focusing on curriculum, Boaler focused her research on the *particular* practices of teaching and learning in the classroom environment, and she examined whether equity can be achieved with low-SES students having teachers who explicitly teach the learning practices necessary for working with open-ended problems. Among the strategies the teachers used were (1) helping students understand what the question demanded and having them restate the problem in their own words, (2) teaching students to see the value for communicating and justification in writing, and (3) discussing with them ways of interpreting questions in context. Results at the end of three years on assessments that included a national exam showed that SBS students not only outperformed the traditional control students but also scored higher than the national average. In addition, whereas boys scored higher than girls did in the traditional group, there were no gender disparities in the SBS group. Thus, Boaler’s results are in direct contrast to the idea that for low-SES students, “the algorithmic mode of instruction might provide a relatively level playing field” (Lubienski, 2000, p. 478). Boaler (2002) recommends that studies involving relational analysis of equity not stop at the curriculum but extend to the teaching and learning practices of teachers, which she perceives as central to the attainment of equality (239).

We can conclude from the research that, given a Standards-based curriculum *and* teachers who can implement it through the CCSSM mathematical practices, the mathematical understanding of students will grow because they now have the “opportunity to learn” (OTL). Boaler (2015), in her book on mathematical mindsets, describes OTL: “Put simply, if students spend time in classes where they are given access to high-level content, they achieve at higher levels” (111). Students not afforded OTL are experiencing *inequality* in instructional content, and, according to Schmidt and Burroughs (2013), it is *more* common in middle-income districts than those in high or low SES, contrary to what one might think (56). More results from their research are in Chapter 12.

In her article examining the achievement gap, Lubienski (2007) provides a valuable perspective on what equity should mean:

If we are truly committed to equitable outcomes, then we must commit *more* resources to those students who most need them. To close achievement gaps in mathematics, we need to ensure that low-SES and minority students get the best teachers, the richest mathematics cur- riculums, the smallest class sizes, and the most careful guidance. Although we might strive to achieve “mathematical power for all,” we will not reach this goal if we focus on all students generally instead of addressing the particular barriers that historically underserved students face in learning mathematics.

*(1)*

Some of the notable successful SB equitable programs in operation today include Project SEED, the Emerging Scholars program, and The Algebra Project. Project Seed, out of Berkeley, California, focuses instruction for low-achieving students in grades 2-8 to prepare them for success in high school mathematics. Rather than continue to try traditional methods to help these students master troublesome topics, Project Seed introduces students to a challenging subject, like algebra, in a way that builds their content knowledge so that they can then revisit the troublesome areas with confidence. Their results have shown that the programs raise students’ test scores by building algebraic and critical thinking and reinforcing basic skills (http://project- seed.org/about-us/). The Emerging Scholars program originated in 1977 as the Mathematics Workshop of the Professional Development Program (PDP) at the University of California at Berkeley It is founded on the research interest of Uri Treisman, then a professor at Berkeley, on why black and Latino students with the potential to finish a college calculus course did not. Observing that black students did not study together, as did their white and Asian counterparts, Treisman and colleagues designed a mathematically rich set of calculus problems to encourage a collaborative work setting. In addition to collaboration, within the PDP program, students found a welcoming and safe space to brainstorm and think creatively The program’s assessments’ result showed two-thirds of the students participating in the Mathematics Workshop at UC Berkeley earning grades of A or B with no students failing (Asera, 2001). Today, Emerging Scholars is open to all interested students and is now widely disseminated across the United States as a part of freshman courses in academic departments. The Algebra Project, founded by Robert Moses, helps African American children develop a conceptual understanding of algebra by linking experiences they intuitively understand to algebraic thinking. Further detail about the project is in Lynne Godfrey’s profile.