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Balance the Equation - Glossary of Terms

In order to facilitate a mutual understanding of terminology used throughout this challenge, definitions have been provided by the Gates Foundation based on research and intention.

Algebra 1: The course is historically situated between pre-Algebra and Geometry and the last truly general-purpose-college/career-ready mathematics course that most students are exposed to in the 8th or 9th grade. “Algebra moves students beyond an emphasis on arithmetic operations to focus on the use of symbols to represent numbers and express mathematical relationships.[1] Algebra provides the language in which we communicate the patterns in mathematics so its foundation for students (including English Learners) is critical for continuing mathematical comprehension. Each topic within Algebra should be experienced as an integration of procedures, concepts, and applications. Algebra typically covers: seeing structure in expressions, arithmetic with polynomials and rational functions, creating equations, and reasoning with equations and inequalities.[2]

Assets: Strengths and interests that a student possesses and brings to the mathematics classroom. Assets can include skills, knowledge, connections/relationships, cultures, dreams, passions, etc.

Critical consciousness: We adopt the definition from the Mindset Scholars Network: educators must understand how marginalization and bias are expressed in mathematics environments and work to actively counter these processes via their instructional choices and interactions with students. Examples of what educator critical consciousness can look in practice include: confronting microaggressions, employing complex instruction, explicitly praising the contributions of students who have a minoritized identity in mathematics, or incorporating students’ uses of mathematics outside of school into their classwork. Learn more at Mindset Scholars Network.

Designed for use together with a high-quality core curriculum: This opportunity envisions developing innovative supplemental resources that can be used as a support that expands access to core content while addressing a variety of student learning needs, and harnessing students’ identities, interests, and creativity. To maximize student impact and expand access to challenging content that is too often withheld from priority students, successful proposals will identify a specific core mathematics curriculum or course with which the new solutions are meant to be paired. Core curriculum may be a standalone Algebra 1 course, or a middle school mathematics curriculum that includes algebraic concepts in 7th and 8th grades. Solutions should set forth clear use cases: (e.g., differentiation for group or individual practice; formal intervention), including potential setting (e.g., in general education classroom; in intervention settings; for use at home leveraging virtual supports).

Effective mathematics instructional practices: The IES Practice Guide Teaching Strategies for Improving Algebra in Middle School and High School Students offers recommendations. In addition, we emphasize:

  • Increased ability to develop an inclusive learning environment in mathematics.
  • Increased ability to support students in deeply engaging with the content (i.e., through discussion, connecting to their lives, problem solving, etc.)
  • Increased ability to assess understanding and mathematics processes (i.e., problem solving.)

Engagement: Students that currently hold a growth mindset about their mathematical abilities, actively seek challenges, exhibit a willingness and confidence to participate in-class or help other classmates work through tasks, have extended periods of uninterrupted focus on a task or instruction, or have an enduring interest in pursuing mathematics or mathematics-related fields based on their academic goals or future aspirations despite poor instructional practices from unsophisticated educators, low-quality teaching materials, disruptive learning environments, emotional blockages, and systemic assessment pressures.

English Learners: Federally defined, "English Learners (ELs)," or Emerging Multilinguals, are students from homes where languages other than English are primarily spoken. ELs are a rich and heterogeneous group of learners who bring significant intellectual and cultural assets to the mathematics classroom, which is not always designed with their success in mind. Providing mathematics content that is designed to build the language of mathematics will be important to serve the nation's growing population of EL students, in ways that are culturally relevant to their linguistic assets as well as mathematically rigorous and appropriate per grade level. Please see the Migration Policy Institute or ELSF for further information.

Equity: Universal exposure so individual student goals can be achieved. "This requires all stakeholders:

  • Ensure that all students have access to a challenging mathematics curriculum, taught by skilled and effective teachers who differentiate instruction as needed;
  • Monitor student progress and make needed accommodations; and
  • Offer remediation or additional challenges when appropriate."[3]

High-quality curriculum: Evaluated and ranked in the top tier from a third-party reviewer based on a set of rigorous standards and alignment for instructional materials. Examples of reviewers include: EdReports, Instructional Materials Evaluation Toolkit (IMET), or Equip rubrics (K8 math units).

Inclusive mathematics communities: Foster a sense of belonging and help all students to develop their identities as competent and capable learners, and to feel a sense of cultural continuity in that context. Inclusive mathematics communities recognize that some student identities have been marginalized within mathematics and actively work to move to deeper inclusion and value of all students as mathematicians. Curricular materials, assessment practices, and classroom interactions contribute to each student's experience of a mathematics community as inclusive or exclusionary. For more details, view Mindset Scholars Network's Five Guiding Principles for Creating Inclusive Mathematics Environments.

Mathematical proficiency: We adopt the National Academies’ field consensus from Adding It Up (2001), on what it means for a student to be proficient. Mathematical proficiency has five strands:

  • Adaptive reasoning: capacity for logical thought, reflection, explanation, and justification
  • Conceptual understanding: comprehension of mathematical concepts, operations, and relations
  • Procedural fluency: skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
  • Productive disposition: habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy
  • Strategic competence: ability to formulate, represent, and solve mathematical problems

Minority-led organization (MLO): We are committed to increasing the presence of MLOs in our investment portfolio. We believe that organizations that draw on diversity in the leadership and staff structures are well-equipped to serve a U.S. public school population that continues to become more ethnically and racially diverse. We define MLOs as those that meet one or more of the following criteria:

  • An organizational leader (e.g., Superintendent, Executive Director, President, or Chief Executive Officer) identifying as Black and/or Latino.
  • An executive leadership team (e.g., center directors, CFO) in which at least 40% of the members identify as Black and/or Latino.
  • A Board of Directors in which at least 40% of the members identify as Black and/or Latino.
  • An organization with programmatic staff (i.e., full-time staff members that make programmatic design and implementation decisions) of which at least 40% identify as Black and/or Latino.

Positive mathematics identity: Mathematics identity specifically relates to an individual’s sense of being a "mathematics person," feeling empowered to engage in mathematics.[4] Through this challenge, we also endeavor for students to understand Algebra's relevance to their lives today and in the future.

Positive experience in mathematics classrooms: This Grand Challenge seeks to increase the positive experiences our priority students are having in mathematics classrooms. Four key outcomes for students include:

  • Increased sense of belonging
  • Increased engagement in Algebra
  • Increased enjoyment in Algebra
  • Increased experiences that encourage deep mathematical thinking, exploration, and collaboration

Positive mindsets/beliefs about priority student mathematics learning: Positive mindsets/beliefs about priority student mathematics learning can take a variety of forms, but three key outcomes for educators/adults include:

  • Increased critical consciousness (understand how marginalization and bias are expressed in mathematics environments and work to actively counter these processes via their instructional choices and interactions with students)
  • Increased belief in priority students as mathematicians (therefore providing access to rigorous mathematics content and pathways)
  • Increased view on the expansiveness of mathematics (i.e.: its history, participants, application and therefore how to increase relevance to priority students' lives)

Priority students: Black, Latino/a, English learners, and/or students experiencing poverty in the United States.

Solution: A solution can be a program, practice, instructional model, platform, or tool supporting priority students (and their teachers) in Algebra 1 in grades 7-9. A solution could be a new solution (that does not exist) or it could be an existing, "operable" solution. A solution could also be built through partnerships with multiple organizations. A solution must align to at least one Area of Focus and be usable explicitly with a full course Algebra 1 program that meets minimum requirements for coverage of algebraic content in grades 7-9. All solutions must be designed to support access to mastery by the end of 9th grade of the content standards typically associated with Algebra 1 courses. We also encourage you to reflect on what other components of a coherent mathematics instructional system (independent practice, intervention, and assessment to inform instruction), your solution could cover. Examples of solutions might include:

  • An online independent practice tool that pairs with an existing high-quality Algebra 1 core curricula, with culturally responsive education and embedded English Learner supports.
  • An after school, summer, or community-based program focused on building positive mathematics identity in Black or Latina girls in Algebra 1, in ways that tie back to the Algebra work happening in the classroom.
  • An in-school tutoring intervention for 7-9th grade students that supports priority students in mathematics through both mentoring and academic rigor.

For more solution ideas, please refer to the earlier section, Why Balance the Equation, Why Now? As you innovate and create your new or existing solution we encourage you to have an eye towards (1) scalable innovations that could be equitably implemented and (2) innovations that will support systems change. If selected for Phase 2, your solution would need to be ready to be implemented with priority students by Fall 2021.

Traditional mathematics classroom: The integrated sum of our inherited and internalized assumptions, values, and beliefs of what it means to be 'good at mathematics' – a "naturally" talented individual as opposed to a collective of adept and dedicated problem solvers, quickly completing arithmetic problems through a procedural orientation (acquisition of a skill through repetition of tasks and practice) and working towards a binary right or wrong solution.



[1] "Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students," July, 21, 2020, https://ies.ed.gov/ncee/wwc/Docs/practiceguide/wwc_algebra_summary_072115.pdf

[2] "High School: Algebra » Introduction," September, 2020, http://www.corestandards.org/Math/Content/HSA/introduction/

[3] "Access and Equity in Mathematics," April 18, 2014, https://www.nctm.org/Standards-and-Positions/Position-Statements/Access-and-Equity-in-Mathematics-Education/

[4] Miller-Cotto, Dana and Lewis, Neil A. (2020). Am I a "Math Person"? How Classroom Cultures Shape Math Identity Among Black and Latinx Students. Working Paper.

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