James C. Kennedy Institute for Educational Success
Position Paper #3
Equity from an Assets-Based Perspective
Richard Kitchen
Spring/Summer 2015
In this paper, I focus on what it means to view equity in mathematics education
from an asset perspective. Examples are provided to elucidate the importance of eliciting
students’ mathematical ideas and constructing instruction based on these ideas. I will also
provide an example that demonstrates the importance of students who have historically
been marginalized in mathematics education in the United States (U.S.), students of color
and low-income students, experiencing mathematical success.
The achievement gap in mathematics has become a taken-for-granted aspect of
the educational landscape in the U.S. (Gutiérrez, 2008). However, instead of accepting
the achievement gap as innate and immutable to change, it can be viewed as an
“opportunity gap” (Flores, 2008). From this perspective, the access that students have to
opportunities to learn mathematics is at the root of differences in student achievement in
mathematics. Student access to a challenging standards-based mathematics education1 is
influenced by race, ethnicity, socioeconomic status (SES), and English language
proficiency (Gutiérrez, 2008; Kitchen, DePree, Celedón-Pattichis, & Brinkerhoff, 2007;
Martin, 2013). For instance, schools that enroll large numbers of African American
students often have disproportionally high numbers of remedial classes in mathematics in
1
Standards-based reforms in mathematics refer to mathematics curriculum and
instruction that promote the development of student reasoning through problem solving
and discourse (see for example, NCTM, 1989; 2000; NSF, 1996).
1 which instruction is focused on rote-learning and strategies that are intended to help
students be successful on standardized tests (Davis & Martin, 2008; Lattimore, 2005).
Brynes and Miller (2007) found that SES has direct effects on mathematics
achievement because students from higher SES homes have access to better-trained
teachers, among other things, and tend to perform at higher levels than students from
lower SES backgrounds. Moreover, SES has indirect effects on both the opportunities
students have to enroll in advanced mathematics classes in high school (i.e., students in
lower SES communities tend to have less access to advanced mathematics courses) and
on their propensity to take advantage of learning opportunities in mathematics (Brynes &
Miller, 2007). Hogrebe and Tate (2012) found that algebra performance is influenced by
where students live; the SES of local communities is significantly related to students’
performance in algebra (i.e., the higher the SES of the community, the higher the algebra
performance). These are just a few examples of how students in low-income communities
do not have the same opportunities as their counterparts from higher SES communities to
engage in standards-based mathematics curriculum and instruction (Kitchen, 2003;
Martin, 2013).
In addition to poverty and SES, student access to a challenging standards-based
mathematics education is influenced by race, ethnicity, and English language proficiency
(DiME, 2007; Gutiérrez, 2008; Martin, 2013). A role that mathematics has played
historically is to sort and stratify students by race, ethnicity, and gender (DiME, 2007;
Gerdes, 1988). Specifically, white and Asian middle class and upper-middle class
students have been privileged to have greater access to challenging mathematics
curriculum and instruction (DiME, 2007; Tate, 1995). Schools that enroll large numbers
2 of African American students often have disproportionally high numbers of remedial
classes in mathematics in which instruction is focused on rote-learning and strategies that
are intended to help students be successful on standardized tests (Davis & Martin, 2008;
Lattimore, 2005). In response to NCLB (2001) and the demands to increase test scores,
Davis and Martin (2008) argue that the preponderance of skills based instruction
“[negatively] shape the lives of poor African American students in more significant ways
than middle-class or affluent students” (p. 18).
In schools that serve large numbers of immigrant Latino/a students who speak
with an accent, use English words incorrectly or speak in Spanish as a means to express
themselves, educators, peers and community members may assume students lack the
capacity to perform well in mathematics (Moll & Ruiz, 2002; Téllez, Moschkovich, &
Civil, 2011). “Deficit perspectives” such as these attribute lower levels of academic
achievement to specific ethnic/racial groups based upon characteristics such as lack of
fluency in English, life experiences that do not parallel those of the dominant society, or
low family income (Gutiérrez, 2008; Spielman, & Mistele, 2013).
Instead of looking at students and their communities through a deficit lens,
students can be viewed as having funds of knowledge such as knowing one language and
learning another, having experiences that are richly grounded in their culture, and having
extensive mathematics experiences in their daily lives (Moll & Ruiz, 2002). If educators
build on the attributes students possess and treat them as mathematically competent, there
is greater potential for increased academic success and an enhanced mathematical
identity (Kitchen, Burr, & Castellón, 2010; Turner, Celedón-Pattichis, & Marshall, 2008).
In the first vignette that followings, Marisol demonstrates that she has mathematical ideas
3 that the instructor can use to help her arrive at a solution to the problem presented.
Throughout, the instructor her ideas are taken seriously by the instructor, rather than
dismissed as not correct.
Vignette #1: Building on ELL Students’ Mathematical Ideas in Instruction
In the following vignette, a description is provided of how Marisol solved the
following problem:
!
!
Veronica has 5 ! pounds of grapes. She gave 2 ! pounds to Marisol. How many
pounds of grapes does Veronica have left?
Initially, Marisol subtracted the whole numbers; she then converted the fractional
!
!
!
!
portion of the mixed numbers ! and !, to !" and !", respectively. She indicated that she
!
!
was not sure how to compute !" − !", because she did not believe it to be possible to take
eight from three. So, Marisol decided to subtract three from eight and got six due to a
computational error. She then inserted a negative sign in front of the number, arriving at
!!
!"
.
The interviewer proceeded by asked Marisol how she could combine three and the
fractional part of the solution,
!!
!"
. She indicated that the six (the numerator of the
4 fraction) could borrow something from the three. To represent the three wholes, she drew
three rectangles and divided each one into 12 pieces. She then drew an extra rectangle to
!
represent !":
3 −6
12
Marisol spent several minutes thinking about what to do next, and decided to leave the
!!
answer as 3 !" . She stated, “I do not like negatives.”
During the next stage of the interview, Marisol explained the same procedure to
solve the problem just described, but realized that she made a mistake when subtracting
!!
the fractions and decided to change her solution to 3 !" . The interviewer questioned her
about the meaning of the negative fraction within the mixed number and asked if the
actual solution is less than or greater than three. Marisol was able to recognize that this
value would be less than three. In the end, Marisol did not combine the whole number
and the negative fraction.
Though Marisol did not arrive at what may be considered a legitimate solution,
she clearly demonstrated correct mathematical thinking. Working in a learning
environment in which her ideas were actively sought out and respected, Marisol created a
strategy to solve the problem. This strategy appeared to be of her own making, that is, she
was not applying a learned algorithm to derive a solution. Precisely because she felt
encouraged to explore her ideas and was treated as mathematically competent, Marisol
took the chance of deriving a solution in her own way. The learning environment that had
been created for Marisol paved the way for her to experiment and pursue a problem
5 solving strategy that made sense to her. Without a doubt, she would have been penalized
!!
for arriving at a solution of 3 !" on a paper/pencil examination, and would mostly likely
!!
have been penalized for the solution 3 !" .
In this vignette, Marisol is treated as mathematically competent (Kitchen, Burr, &
Castellón, 2010; Turner, Celedón-Pattichis, & Marshall, 2008), and she demonstrates that
she has sound mathematical ideas. As Marisol’s teacher, it was clear to me from this
example that she needed help understanding negative numbers. It was also important to
!!
!
!
assist her to understand that 3 !" can also be represented as ! − !". She demonstrated a
strategy that made sense to her as a means to derive a solution. As her instructor, it was
important to me to both validate this strategy and look for opportunities for Marisol to
show and explain this strategy to her peers as a valid way to solve problems involving the
addition and subtraction of mixed numbers. Through validating Marisol’s ideas, I could
help her develop a positive sense of herself in mathematics, that is, enhance her
mathematical identity (Boaler, 2002; Cobb, Gresalfi, & Hodge, 2009; Martin, 2000; Nasir
& Hand, 2008). In this next vignette, some examples are provided to demonstrate the
importance of students experiencing mathematical success (Kitchen, 2015).
6 Vignette #2: The Importance of Students Experiencing Mathematical Success
For the duration of the 2007-08 school year, I taught a 6th grade class using the
Connected Mathematics Program 2 (CMP) in Albuquerque, New Mexico. I had 17
students in the class, 13 girls and 4 boys. At the time, all of the students would have
qualified for Free Reduced Lunch. The class composition was also highly diverse; 14 of
the students were of Mexican descent and 12 were English language learners (ELLs).
Two of the students are twin sisters and African-American. For the duration of the year, I
maintained a journal of my experiences. In this vignette, I share what I learned about the
importance of students, particularly low-income students and racial/ethnic minority
students2, having the time and support needed to learn challenging mathematical ideas.
A particularly important revelation for me as a 6th grade CMP teacher concerned
the fragility of many of my students’ “math egos.” It should not be underestimated how
tentative middle school students may feel about themselves and their mathematical
abilities. It is important to note that many of my students who exhibited among the most
fragile math egos were my “struggling learners,” these students had not experienced
much success in mathematics. Importantly, many of these students are students of color. I
can imagine many students with fragile math egos giving up quickly in CMP classrooms
if they do not have regular access to teachers and tutors who can help them make sense of
mathematical ideas.
What can CMP teachers do to help students develop a positive mathematical
identity? What I learned is that students, particularly students with fragile math egos,
2
I use “low-income students” as synonymous with students who are classified as living in
poverty in the United States (U.S. Census, 2010). “Students of color” is used as
synonymous with “ethnic and racial minority students” and “culturally and linguistically
diverse students.”
7 need to continually experience success in mathematics. Success for some students may be
as simple as finding the product of two single digit numbers. This is especially the case
with learners who have not experienced much prior success in their mathematics classes.
To support student success and even celebrate them, I looked for occasions to emphasize
my 6th graders’ brilliance (Martin & Leonard, 2013). For example, during one unit, I
made a point to continually call on Britney, one of the African American twins, because
she was doing a great job of talking the class through how to multiply two 2-digit
numbers. She became our class expert on 2-digit multiplication; something that should
took pride in as the year progress. During another lesson, I continually called on Janie, an
ELL who was among my weakest students in mathematics, but who was doing a nice job
of filling in the blanks in the multiplying decimal activities (e.g., 2.4 x 10 = 24.0,
therefore .24 x 10 = __). My goal in both cases was to purposely highlight the
accomplishments of Britney and Janie, to position them as mathematically capable
(Kitchen, Burr, & Castellón, 2010; Turner, Celedón-Pattichis, & Marshall, 2008), and to
support their sense of themselves as able learners of mathematics. It is also important to
note that both students are female students of color. In my effort to support the
development of positive student mathematical identity, I believe it is vital to intentionally
notice and validate the mathematical thinking and accomplishments of historically
marginalized student groups in mathematics (Martin, 2000).
I also learned that reinforcing my students’ mathematical learning through
practice helped support their emerging positive mathematics identities. As students were
learning important mathematical concepts, they needed time to reflect upon what they
had learned through exploration and to solidify what they had learned through practice.
8 For example, as described earlier, we engaged in excellent investigations that helped
students make sense of how to find decimal products. After participating in some nice
explorations such as 24 x 10 = 240, find 2.4 x 10, students needed time to find the
products of other decimal numbers to solidify their understanding and continue to gain
confidence in their newfound knowledge. This proved particularly important for students
with fragile math egos.
Final Remarks
Supporting students to have rich opportunities to engage in, have success with,
and potentially even relish learning challenging mathematical content is demanding. A
tragic educational legacy in the U.S. is that many students, particularly low-income
students of color, have had limited access to opportunities to engage in a challenging
standards-based mathematics education (Kitchen, et al., 2007; Martin, 2013). In the
vignette provided about Marisol and in my work to engage 6th graders, I learned about the
importance of validating the mathematical ideas that students bring to the classroom and
supporting students to have success with mathematics. My students also needed to just
practice some of the complex ideas they had devoted significant effort to make sense of,
9 and success through practice helped support their developing a positive sense of
themselves and what they are capable of doing in mathematics. To be clear, I am not
advocating here for practice that equates with the sort of low-level mathematics education
that has historically been found in schools that primarily serve low-income, students of
color in which the memorization of math facts, algorithms, vocabulary and procedures
are the focal point of instruction, rather than teaching students through the use of
complex, challenging problems (Davis & Martin, 2008; Kitchen, et al., 2007; Lattimore,
2005). My point is simply that practice can play a role when a standards-based
mathematics program is in use to support students’ burgeoning mathematical identities.
Specifically, I found practice helped my students, particularly those who had previously
experienced little success in mathematics, gain confidence in their abilities and begin to
believe that they could in fact do and learn some mathematics.
Interestingly, experts on Response to Intervention (RtI), a program that is being
used nationally to provide early interventions for students who experience learning
difficulties (Fuchs, Vaughn, & Fuchs, 2008) recognize that mathematical success leads to
more success for struggling students (Formative Assessment Working Meeting, 2014). I
believe that we need to focus more in the mathematics education community on
providing opportunities for students, particularly students who have not had many
uplifting experiences in mathematics, to have more mathematical successes. This is
especially important if a teacher is using a challenging, standards-based mathematics
program in a classroom with students who may not possess positive identities about
themselves in mathematics.
10 References
Boaler, J. (2002). The development of disciplinary relationships: Knowledge, practice,
and identity in mathematics classrooms. For the Learning of Mathematics, 22(1),
42-47.
Brynes, J. P., & Miller, D. C. (2007). The relative importance of predictors of math and science
achievement: An opportunity-propensity analysis. Contemporary Educational
Psychology, 32, 599-629.
Cobb, P., Gresalfi, M., & Hodge, L.L. (2009). An interpretive scheme for analyzing the
identities that students develop in mathematics classrooms. Journal for Research
in Mathematics Education, 40(1), 40-68.
Davis, J., & Martin, D. B. (2008). Racism, assessment, and instructional practices:
Implications for mathematics teachers of African American students. Journal of
Urban Mathematics Education, 1(1), 10-34.
Diversity in Mathematics Education (DiME) Center for Learning and Teaching. (2007). Culture,
race, power and mathematics education. In F. K. Lester (Ed.), Second handbook of
research on mathematics teaching and learning (pp. 405–433). Charlotte, NC:
Information Age.
Flores, A. (2008). The opportunity gap. In R. S. Kitchen, & E. Silver (Editors),
Promoting high participation and success in mathematics by Hispanic students:
Examining opportunities and probing promising practices [A Research
Monograph of TODOS: Mathematics for All], 1, 1-18. Washington, DC: National
Education Association.
11 Formative Assessment Working Meeting, (2014). School of Education, University of
Michigan, Ann Arbor, MI, October 12-14.
Fuchs, D., Vaughn, S.R., & Fuchs, L.S. (Eds.). (2008). Responsiveness to intervention: A
framework for reading educators. Newark, DE: International Reading
Association.
Gerdes, P. (1988). On culture, geometrical thinking and mathematics education. Educational
Studies in Mathematics 19, 137-162.
Gutiérrez, R. (2008). A “gap gazing” fetish in mathematics education? Problematizing
research on the achievement gap. Journal for Research in Mathematics Education,
39, 357–364.
Hogrebe, M. C., & Tate, W. F. (2012). Place, poverty, and Algebra: A statewide
comparative spatial analysis of variable relationships. Journal of Mathematics
Education at Teachers College, 3, 12-24.
Kitchen, R. S. (2003). Getting real about mathematics education reform in high poverty
communities. For the Learning of Mathematics, 23(3), 16-22.
Kitchen, R.S. (2015, Winter). Supporting the success of diverse, low-income learners in a
Connected Mathematics Program (CMP) class. Colorado Mathematics Teacher,
22-26.
Kitchen, R. S., Burr, L., & Castellón, L. B. (2010). Cultivating a culturally affirming and
empowering learning environment for Latino/a youth through formative
assessment. In R. S. Kitchen, & E. Silver (Editors), Assessing English language
learners in mathematics [A Research Monograph of TODOS: Mathematics for
All], 2, 59-82. Washington, DC: National Education Association.
12 Kitchen, R. S., DePree, J., Celedón-Pattichis, S., & Brinkerhoff, J. (2007). Mathematics
education at highly effective schools that serve the poor: Strategies for change.
Mahwah, NJ: Lawrence Erlbaum Associates.
Lattimore, R. (2005). African American students’ perceptions of their preparation for a
high-stakes mathematics test. The Negro Educational Review, 56 (2 & 3), 135–
146.
Martin, D. (2000). Mathematics success and failure among African-American youth: The
roles of sociohistorical context, community forces, school influence, and
individual agency. Mahwah, NJ: Lawrence Erlbaum Associates.
Martin, D. B. (2013). Race, racial projects, and mathematics education. Journal for
Research in Mathematics Education, 44, 316–333.
Martin, D. B. & Leonard, J. (2013). Beyond the numbers and toward new discourse: The
brilliance of Black children in mathematics. Charlotte, NC: Information Age
Publishing.
Moll, L. C., & Ruiz, R. (2002). The schooling of Latino children. In M. M. SuárezOrozco & M. M. Páez, (Eds.), Latinos: Remaking America, (pp. 362-374).
Berkley, CA: University of California Press.
Nasir, N. S., & Hand, V. (2008). From the court to the classroom: Opportunities for
engagement, learning and identity in basketball and classroom mathematics.
Journal of the Learning Sciences, 17(2), 143-180.
No Child Left Behind Act of 2001. (2001). Retrieved at
http://www.ed.gov/policy/elsec/leg/esea02/index.html.
13 Spielman, L. J., & Mistele, J. (Eds.). (2013). Mathematics teacher education in the public
interest: Equity and social justice. Charlotte, NC: Information Age Publishing.
Tate, W. F. (1995). Economics, equity, and the national mathematics assessment: Are we
creating a national tollroad? In W.G. Secada, E. Fennema, & L.B. Adajian (Eds.), New
directions for equity in mathematics education (pp. 191-208). New York: Cambridge
University Press.
Téllez, K., Moschkovich, J. N., & Civil, M. (Eds.). (2011). Latinos and mathematics
education: Research on learning and teaching in classrooms and communities.
Charlotte, NC: Information Age Publishing.
Turner, E., Celedón-Pattichis, S., Marshall, M., (2008). Cultural and linguistic resources
to promote problem solving and mathematical discourse among Hispanic
kindergarten students. In R. S. Kitchen, & E. Silver (Editors), Promoting high
participation and success in mathematics by Hispanic students: Examining
opportunities and probing promising practices [A Research Monograph of
TODOS: Mathematics for All], 1, 19-40. Washington, DC: National Education
Association.
U.S. Census. (2010). How the Census Bureau measures poverty. Retrieved from
<http://www.census.gov/hhes/www/poverty/about/overview/measure.html>.
14