Academic Quarterly

Marvelous Mathematician Spring 2021

Address Unfinished Learning by Focusing on the Topics That Are Absolutely Necessary

Addressing unfinished teaching and learning — required material from prior grade levels that students have not yet mastered — is expected to be among the greatest challenges going into the next school year (Dorn et al. 2020). Much of this missing knowledge is necessary for ongoing grade-level work. For that reason, we must address this need systematically and sufficiently. It cannot be something we simply assign to teachers to solve on their own through bits and pieces of just-in-time learning, some scaffolding, some small group work, and some differentiation. To do so is to continue a major inequity in our schools — that of failing to provide the support students need for success.

Gaps in knowledge come in many different sizes. Some, we can just step over and kids will get it as we work on the grade-level content. Other gaps we can throw a few two-by-fours over and walk across while making the connections between prior learning and new learning explicit (using just-in-time teaching, scaffolding, and differentiation). Other gaps will require building a complete bridge to connect where students are to where they need to be for success with grade-level work. Our goal is not to completely fill in the entire gap. We cannot try to reteach all of last year’s curriculum or, for goodness’ sake, all of the last two- or three-years’ curricula. However, we can target the most important prerequisites to give students the best chance for success with the math content they need to learn. The essential questions are “What prior knowledge should we focus on? Which students need this instruction? How should we best address unfinished learning? What knowledge and abilities do students have that we can leverage?,” and just as importantly, “When should we teach this necessary prerequisite knowledge?”

Not all math content is created equal. There are such things as the “major work of the grade, prerequisite knowledge” and, as described by NCTM and NCSM (2020), “mathematics that is absolutely essential for students to learn” (p. 6). One of NCTM’s and NCSM’s essential questions to be answered in determining what needs to be taught is “Will student learning in future grades and courses be hindered without understanding and proficiency of this standard?” (p. 6). Evidence points to building number sense and fluency with numbers, operations, and solving equations as the most essential prerequisite knowledge from first grade through Algebra. Several recent and older research-based publications provide the evidence for this claim.

In 2008 the National Mathematics Advisory Panel (NMAP) released its Final Reporthighlighting some key needs for success with Algebra. In the report are the following recommendations:

Proficiency with whole numbers, fractions, and particular aspects of geometry and measurement [primarily work with similar triangles] should be understood as the Critical Foundations of Algebra. Emphasis on these essential concepts and skills must be provided at the elementary and middle grade levels.

The coherence and sequential nature of mathematics dictate the foundation skills that are necessary for the learning of algebra. The most important foundational skill not presently developed appears to be proficiency with fractions (including decimals, percent, and negative fractions). The teaching of fractions must be acknowledged as critically important and improved before an increase in student achievement in algebra can be expected. (p. 18)

In its findings from research, the panel concluded that number sense and fluency with whole numbers and rational numbers are essential for successful algebra learning and mastery.

In the Institute of Education Sciences (IES) Practice Guide Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools (2009), specific areas of mathematics that are identified as needing support are problem-solving and fluency with number facts. The IES reinforces these recommendations (while adding some others, such as mathematical language) in its most recent publication focused on intervention for elementary grades in mathematics (2021). Do not consider this a call for rote memorization. Fluency and problem-solving are built on a foundation of conceptual understanding. As stated by NMAP (2008), “Conceptual understanding of mathematical operations, fluent execution of procedures, and fast access to number combinations together support effective and efficient problem solving” (p. 26). NMAP further states that “poor number sense interferes with learning algorithms and number facts and prevents use of strategies to verify if solutions to problems are reasonable” (p. 27). In sum, keys to mathematical growth are solid number sense and fluency.

In June of 2020 the Council of the Great City Schools (CGCS) released Addressing Unfinished Learning After COVID-19 School Closures. In this document CGCS provides recommendations for major focus areas within previous grade bands for success with current grade bands (3–5, 6–8, HS). The report is based on the belief that:

Grade-level work in mathematics makes extensive use of knowledge and skills taught in prior grades. The priorities below identify knowledge and skills from prior grade spans most needed—and most likely to show up as unfinished learning—as students engage in grade-level work. (p. 10)

For transitioning to third grade, five areas of focus are identified. The three largest areas of focus are:

  • Understand the properties of addition (any-which-way) and the relationship between addition and subtraction.
  • Interpret addition subtraction situations (add to, take from, put together and take apart, compare), and express the corresponding additions and subtractions as number equations.
  • Understand the role of place value in counting, addition and subtraction. (pp. 11–12)

For transitioning to sixth grade, six areas of focus are identified. The four largest areas of focus are:

  • Extend whole number arithmetic, the properties of operations and number lines to decimal numbers and fractions.
  • Understand the properties of multiplication, with whole numbers, fractions and decimals.
  • Understand the role of place value in multiplication and division.
  • Interpret multiplication and division situations (sharing and equal groups, area and array, compare). (pp. 13–14)

For transitioning to Algebra 1, six areas of focus are identified. Two of the three largest areas of focus are:

  • Use the properties of operations, properties of equality and general principles of algebra rather than specialized techniques for specialized problems to create, interpret, and solve one-variable linear equations and pairs of simultaneous two-variable linear equations.
  • Extend understanding of numbers, the properties of operations, and the number line to include rational numbers including negative numbers. (pp. 15–17)

Most recommendations from the CGCS call for building proficiency in number sense and fluency with numbers, operations, and solving equations. These are the major areas of focus from prior grade bands to address the likely unfinished learning in current grade bands.

Finally, we look at a 2021 publication from the Achievement Network (ANET) that identifies essential prerequisite math standards from the immediate prior grade level for current grade-level work (through Algebra 1). It is important to note that only prerequisite work from one grade level back is identified for each grade level. However, the document reinforces two key ideas:

  • Some prior knowledge is absolutely necessary for success with other new knowledge (grade-level content). ANET identifies this as “Bridge up,” which means “Without this prior knowledge, students most likely do not have a way to access the grade-level standard” (p. 1).
  • Almost 80% of the bridge up standards are in the areas of number sense and fluency with numbers, operations, and solving equations.

A numerical analysis of the 39 standards that are listed in the bridge-up category reveals that 31 of these are in the domains of Operations and Algebraic Thinking (OA), Numbers and Operations in Base Ten (NBT), and Number and Operations—Fractions (NF) for grades K–5; and in the domains Number System (NS) and Expressions and Equations (EE) for grades 6–8. Considering that there are approximately 200 standards from grades 1 through 8 in the Common Core State Standards for Mathematics, and that of these 200 standards, approximately 55% of them are in the domains listed above, this list of 39 bridge up standards with 31 in the domains mentioned above does narrow the scope of what unfinished learning we need to prioritize. Many students have unfinished learning that reaches past just one prior grade level. Even with that in mind and looking at least at the prior two years of “absolutely necessary” knowledge, this list is still very helpful in narrowing down our focus.

Regardless of how we look at it, the conclusion is the same. The primary areas of need for addressing unfinished learning that are most impactful for success with grade-level content are with building numbers sense and fluency with numbers, operations, and solving equations. It is important to remember that conceptual knowledge remains an essential foundation for fluency. Therefore, instruction must link together conceptual knowledge with procedural skill. It is also important to recognize that understanding mathematical concepts does require that students be engaged in some productive struggle with the math and that mathematical connections are made explicit (Hiebert and Grouws 2007).

We cannot assume all students have unfinished learning needs in these essential areas. Use formative assessment measures to identify which students have needs and the degree of the support needed. Formative assessments come in many forms, such as pre-assessments with units of instruction, formal assessment tools (such as iReady, NMAP, DIBELS Easy CBM Math, STAR, and Aimsweb), exit tickets at the end of lessons, classwork, and student talk in class. It is important to provide all students with support, whether it is to equip them with the tools they need for success with grade-level content or to provide additional deeper challenges.

The topics of how and when to address unfinished learning are worthy of a book or at least another article on their own. Time being a limited resource for instruction means we are challenged with finding the time and knowing how to use it most productively to address learning needs without endangering necessary progress with grade-level work. The Council of the Great City Schools (2020) says it well:

Curriculum leaders will need to articulate the district’s instructional priorities for schools and teachers—what is most important to teach within the major curricular domains at each grade level. It is important that teachers know where to invest their time and effort, what areas can be cut, and where they should teach only to awareness level to save time for priorities. What is most important deserves more time, and teachers need to be given the latitude to provide responsive feedback and allow time for constructive struggle—a very different proposition than merely offering a superficial ‘right’ or ‘wrong.’ This additional time has to come from somewhere. (p. 4)

CGCS provides an excellent list of recommendations for four levels of support to provide students:

  • For students whose struggle with grade-level assignments and language demands becomes unproductive:
    • Provide extra feedback on student work and pose questions that push students to articulate their thinking and to compare solutions and strategies. Teachers should also include explicit summaries in the classroom while working on grade-level tasks. These approaches require that teachers use discussions and questioning to informally determine the need for additional just-in-time support.
  • For students who need specific prior knowledge and language support for a grade-level assignment:
    • Provide just-in-time instruction that does not disrupt the flow of each lesson. This requires that teachers understand how mathematical concepts and skills progress over time so that lessons include time for re-engagement without significantly interrupting the flow of classroom instruction.
  • For students who need a more significant chunk of specific prior knowledge and language support for a grade-level unit:
    • Provide extra teaching of underlying mathematical concepts and skills connected to the grade-level unit using mini-lessons or centers and spend extra time with students providing explicit feedback on their thinking and tutoring (including peer-to-peer tutoring) during each lesson. This requires that teachers engage in prior planning to provide additional just-in-time instruction.
  • For students who need sustained conceptual understanding and language support to stay engaged in grade-level work:
    • Provide supplemental instruction beyond the regular class that will support student’s success with grade-level work. This requires that teachers engage in prior planning to understand how concepts and skills progress over time so that the supplemental instruction explicitly strengthens the foundational concepts and interconnected language functions needed to access grade-level work in mathematics. (pp. 10–11)

It is important to note that for students with the most significant needs, additional instructional time is recommended. Additional instructional time is recommended by many organizations including the CGSC as shown above, by NCTM and NCSM (2020), and by The Education Trust (2021). The challenge lies in where to find it.


It is essential that we address the essential learning students need for success with grade-level work. The idea is not to remove them from grade-level work, but to increase supports as needed to increase students’ likelihood for success with grade-level work and beyond. This is a foundation for equitable instruction. One of the most important pieces of the puzzle is knowing what unfinished learning to address with the limited time we have available to help students. Evidence from research points directly to work in the areas of number sense and fluency with numbers, operations, and solving equations. By maintaining a focus in these areas, we can best assist students in building their mathematical proficiency and succeeding with high-quality math instruction.


Achievement Network. (2021). Important prerequisite math standards. Retrieved from

Council of the Great City Schools. (2020, June). Addressing unfinished learning after COVID-19 school closures. Retrieved from

Dorn, E., Hancock, B., Sarakatsannis, J., and Viruleg, E. (2020). COVID-19 and learning loss—disparities grow and students need help. McKinsey & Company. December 8, 2020. Retrieved from

The Education Trust. (2021). Strategies to solve unfinished learning. Retrieved from

Fuchs, L. S., Bucka, N., Clarke, B., Dougherty, B., Jordan, N. C., Karp, K. S., and Woodward, J. (2021). Assisting students struggling with mathematics: Intervention in the elementary grades (WWC 2021006). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from

Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. (2009). Assisting students struggling with mathematics: Response to intervention (RtI) for elementary and middle schools (NCEE 2009-4060 ed.). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from

Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students learning. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371–404). Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics and National Council of Supervisors of Mathematics. (2020). Moving forward: Mathematics learning in the era of COVID-19. NCTM & NCSM June 2020. Retrieved from

National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education. Retrieved from


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