I am a devoted follower of the late Sir Ken Robinson (may he rest in peace). Somewhere in his massive body of work he described math as a party to which many of us feel we have not been invited. Although I can't find the quote I have never forgotten it because I'm one of the people that got excluded from that guest list--or at least that's what I've always believed. I've gone along with the casual dogma of, "there are math people, and there are non-math people."
In truth, math is not a party at all: it's a ladder.
Unlike other academic content areas, math concepts are cumulative, making any gaps specifically problematic. If I am still struggling to master the concepts taught in 5th grade, I will have little chance of being successful when I'm forced in 6th grade to tackle new concepts. For example, working with percentages is a mystery if you never learned decimals. However, my 6th grade teacher has to teach the 6th grade level math content (regardless of students’ zones of proximal development) because that's what will be tested.
Grade-level accountability systems that are meant to safeguard equity, are, in this case, prohibiting student-centric learning. This is actually a misinterpretation of equity, which doesn't demand that all students receive identical instruction, but rather that "reasonable and appropriate accommodations be made as needed to promote access and attainment for all students" (NCTM 2000, p. 12). And as a result, my learning gaps accumulate.
My ladder was missing two rungs in 5th grade but by the end of 6th grade it's missing four rungs. It becomes increasingly tricky for me to ascend at all. It's not that my brain is incapable of computing and comprehending--I just have learning gaps. Because math is aggregative, those gaps have caused me to stay stuck on the lower rungs of the math ladder my whole life, shrugging my shoulders and saying, "I never got the invitation. Oh well. I can do other things." I watched the same thing happen to my daughter, and to many of my students. The problem is further compounded for students who are also learning English.
We, as educators, are on the hook here. Research indicates that the "math person/non-math person" doctrine is bogus. All students can learn mathematics when they have access to high-quality mathematics instruction and are given sufficient time and support to master a challenging curriculum (Burris, Heubert, & Levin, 2006; Campbell, 1995; Education Trust, 2005; Griffin, Case, & Siegler, 1994; Knapp et al., 1995; Silver & Stein, 1996; Slavin & Lake, 2008; Usiskin, 2007).
Strategies to Bridge the Gaps
Active and Oral Engagement
Carol Gautier, CEO and founder of TeachTransform[KH1], emphasizes that instruction must be not only personalized and recursive, but fun. TeachTransform provides discussion-driven math activities for grades 3-5. Customizable and differentiated exercises combine rigorous thinking with real world applications delivered through adventurous characters. Oral participation increases engagement while reinforcing verbal skills for all students.
“This is not just an ELL issue,” said David J. Francis[KH2], from the University of Houston. “It’s for all students who are academically at risk. Many will benefit from building academic language and background knowledge through oral language.”
Personalize and Differentiate
"Our activities are story based because the narrative of a story pulls in all learners, but then the way those learners engage in the problem and computation varies based on where each learner is at on the continuum of skills and concepts," says Gautier. "Different skill levels are offered with each activity."
The Iceberg Problem[KH3], published by New Classrooms, reported that when balanced assessments are employed to personalize student learning based on gaps and growth rather than grade level content and competencies, overall growth averaged 20 percentile points. They also found that schools that operated within accountability systems that valued learning growth grew 38 percentile points over a three-year period while those focused largely on state proficiency grew only 7 points.
Ensure Equity & Access
Learning gaps in math often fall along racial, economic and linguistic divides. Seventy-one percent of English Language Learners in the U.S. public school system speak Spanish as their first language[KH4]. As a group they show significant gaps in math scores compared to their peers[KH5]. While it has been widely believed that learning math in English is easier for ELLs since numbers and formulas are language agnostic, this doesn't bear out in practice.
While math may be less language dependent than other subjects, the language used is content specific, uncommon, and complex. The National Council for Teachers of Mathematics (NCTM, 2000) emphasizes the need to hire bilingual staff and provide materials in Spanish, because once understood in the first language, skills are retained when instruction shifts to the second language. Recent research has shown that having a bilingual teacher not only helps ELLs comprehend academic material, but it also helps increase attendance and graduation rates[KH6].
"We've always provided all of our materials--both student-facing and teacher facing--in both Spanish and English," says Gautier. Sentence frames included in each activity add additional support for students.
A Concrete Progression
The Journal of Instructional Pedagogies [KH7] recommends the use of visual support to make language and mathematics more comprehensible for all students, but especially ELLs. As students use manipulatives to make sense of mathematical ideas, they are also learning objectives from the NCTM standards related to problem solving, communicating, reasoning, connections, and estimation.
TeachTransform activities incrementally move students from concrete manipulatives, to pictorial representations, to more abstract representations, and eventually to abstract algorithms. Gautier and her team started by creating activities for the hardest to teach skills and concepts and are now moving backward to easier-to-teach concepts. Teachers can easily find and select the concepts and exercises to customize learning for individuals or small groups.
Rigor and High Expectations for All
Traditional tracking practices have consistently disadvantaged groups of students by relegating them to low-level mathematics classes, where they repeat work year after year, fall further and further behind their peers in grade-level courses, and are not exposed to significant mathematical substance or the types of cognitively demanding tasks that lead to higher achievement (Boaler, Wiliam, & Brown, 2000; Schmidt, Cogan, Houang, & McKnight, 2011; Stiff, Johnson, & Akos, 2011; Tate & Rousseau, 2002).
Eugene Garcia and René Gonzalez, researchers from NCTM and U.S. Department of Education[KH8], stress that while completion times and supports need to be increased for ELLs and students with learning gaps, expectations and rigor should never be lowered or compromised. Students are best equipped to meet and exceed math expectations when they are provided with abundant and diverse opportunities for speaking, listening, reading, and writing as their teachers encourage them to take intellectual risks.
Lowering expectations and rigor is a way of withholding the invitation. It communicates to students that the lower rungs of the ladder are where they belong, and the experts testify that that's simply not true.
[KH8]Garcia, Eugene E., and René Gonzalez. “Issues in Systemic Reform for Culturally and Linguistically Diverse Students.” Teachers College Record, 96, no. 3 (1995): 418–31.