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Reviewing Core Instruction: Math

By: Rachel Brown, Ph.D., NCSP

Last week’s blog discussed the importance of core reading instruction. This one will do the same for math. Effective reading instruction has received a great deal of attention from educators in recent decades. More recently, effective math instruction has been addressed by researchers and policy makers. In 2001, the National Research Council published a summary of research about mathematics instruction titled Adding it Up. This summary identified five key strands of mathematics needed for proficiency:

  1. Adaptive reasoning
  2. Strategic competence
  3. Conceptual understanding
  4. Productive disposition
  5. Procedural fluency

The report also called for better integration of mathematics instructional materials and teaching resources. In 2006 President George W. Bush commissioned the National Mathematics Advisory Panel (NMAP) to develop a set of specific instructional guidelines for teachers. In 2008 the NMAP published Foundations for Success, a summary of its recommendations. This report included six primary recommendations for improving mathematics instruction.

  1. Streamline the curriculum to focus on a key set of essential skills;
  2. Use research-based teaching practices that focus on conceptual understanding, procedural fluency, and basic fact automaticity;
  3. Recruit and retain classroom teachers with strong math skills;
  4. Include a balance of student-centered and teacher-centered instruction;
  5. Improve the quality of state and national (e.g., NAEP) math assessments to include more items about algebra, and;
  6. Invest in research about effective math instruction practices.

The report also emphasized that mathematics skills can be learned by all students and should not be thought of as something only a few mathematically inclined students learn.

Many schools have responded to the recent attention to math instruction by adopting a core (e.g., Tier 1) math curriculum. The goal and purpose of having an adopted core math curriculum is to ensure that all students have equal access to effective mathematics instruction. As with reading, not all students will meet math learning goals through the Tier 1 core instruction. For this reason, schools also need to provide increasingly more intensive instructional supports based on students’ needs through Response to Intervention (RTI) practices or a Multi-Tier System of Supports (MTSS). Although research about core math instruction is not as developed as for reading, this blog will provide guidelines about selecting, using, and evaluating core mathematics instruction.

Core Mathematics Instruction

Core instruction includes the materials and methods in place for teaching ALL students daily lessons in the general education classroom. Although research about effective core math instruction is still in progress, some findings have been affirmed across many studies and should be present in any adopted core math curricula. As noted in the NMAP report (2008, p. xvi):

A focused, coherent progression of mathematics learning, with an emphasis on proficiency with key topics, should become the norm in elementary and middle school mathematics curricula. Any approach that continually revisits topics year after year without closure is to be avoided.

This recommendation reflects the need for U.S. math instruction to include less total topics and more time for mastery of selected content. To that end, the NMAP report recommended that K-8 core math instruction incorporate teaching of the essential skills necessary for students to be ready for algebra by high school (e.g., grade 9). Such curricula should have a focus on mastery of skills and not include a “spiral” approach where topics are revisited without the expectation of student mastery.  

In particular, the NMAP included the following benchmarks for critical math skills at specific grade levels (2008, p. 20).

  1. Fluency With Whole Numbers
    • By the end of Grade 3, students should be proficient with the addition and subtraction of whole numbers.
    • By the end of Grade 5, students should be proficient with multiplication and division of whole numbers.
  2. Fluency With Fractions
    • By the end of Grade 4, students should be able to identify and represent fractions and decimals, and compare them on a number line or with other common representations of fractions and decimals.
    • By the end of Grade 5, students should be proficient with comparing fractions and decimals and common percent, and with the addition and subtraction of fractions and decimals.
    • By the end of Grade 6, students should be proficient with multiplication and division of fractions and decimals.
    • By the end of Grade 6, students should be proficient with all operations involving positive and negative integers.
    • By the end of Grade 7, students should be proficient with all operations involving positive and negative fractions.
    • By the end of Grade 7, students should be able to solve problems involving percent, ratio, and rate and extend this work to proportionality.
  3. Geometry and Measurement
    • By the end of Grade 5, students should be able to solve problems involving perimeter and area of triangles and all quadrilaterals having at least one pair of parallel sides (i.e., trapezoids).
    • By the end of Grade 6, students should be able to analyze the properties of two-dimensional shapes and solve problems involving perimeter and area, and analyze the properties of three dimensional shapes and solve problems involving surface area and volume.
    • By the end of Grade 7, students should be familiar with the relationship between similar triangles and the concept of the slope of a line.

Using Core Mathematics Instruction

No core instruction will help students if it is taught incorrectly. For this reason, once a mathematics core has been selected, investing resources in preparing teachers to use it is important. Such preparation should include both initial and ongoing professional development during each school year. An important consideration is what parts of any published core “program” will be used at each grade level. Most published math programs include more content than needed in each classroom. This is because such programs need to include resources for classrooms in every state. Each U.S. state sets its own learning standards so publishers try to include content that will work for all states. Not all of the materials and lessons included in published programs will be necessary for all teachers and students. Prior to using a new core math program, it is helpful if a local team develops a “pacing guide” which outlines the required lessons and content to be used at each grade level. Such pacing guides help teachers select and use the lessons corresponding to the local and state math learning goals. Such pacing guides assist teachers in following the NMAP guidance that “less is more” when it comes to math instruction.

In addition to a pacing guide, procedures to ensure teaching integrity are needed. Teaching integrity refers to the extent to which the math lessons are provided in the manner that was validated in research studies. There are different approaches to confirming teaching integrity, including self-review, walkthroughs, and instructional tours. Self-review is the easiest method because it involves having the teachers record what lesson steps they did or did not complete during a daily lesson. This method can be a good starting point for confirming integrity but the data might include teacher bias or lack of accurate recall. For this reason, additional teaching integrity review methods might be helpful, especially during the first year of a new core mathematics program. For more information about other teaching integrity review options see the blog about core reading instruction.

Evaluating Core Mathematics Instruction

In order to know which students need additional instruction beyond the math core to reach learning goals, some form of universal screening is important. Such screening involves having all students in a grade complete the same assessment several times a year. FastBridge Learning® offers universal math assessments for grades K-8, including earlyMath, aMath, and several types of CBMmath. Screening data provide two important types of information: (a) how well the core instruction is meeting students’ needs, and (b) which students need additional assistance. When the screening scores indicate that 80% or more of students are meeting the learning goals (e.g., benchmarks or standards) the core instruction is generally understood to be highly effective. When less than 80% of students have scores that meet the goal, steps to strengthen the core math instruction are recommended.

It is important to note that in cases where core math instruction needs to be strengthened, additional efforts to assist those students whose math scores are the weakest are still necessary. In other words, when using the 80% rule to review students’ math performance, teachers must pay attention to the needs of ALL students. For example, if only 60% of the students in a class meet the math benchmark, there is a need to strengthen the math core instruction. Nonetheless, in this situation, there is also a need to provide additional instruction for those students whose math performance is lowest. This is because public schools have a responsibility to meet the learning needs of all students. Those students with the lowest scores should be provided with additional instruction (also called intervention) that focuses on grade level math goals. Such intervention should occur alongside steps to improve outcomes from the core math instruction. In some cases, ongoing data will indicate that the adopted core program does not meet the needs of at least 80% of students. When this is the case, a new core math curriculum should be selected.  

Although mathematics instruction has not received the same amount of attention as reading, key findings from major studies do provide important guidance about Tier 1 core math instruction. A synthesis of available research shows that U.S. math instruction should include a smaller number of topics and that these should be taught in ways that result in student mastery. In order to know if core math instruction is meeting the goal of having at least 80% of students reach the grade level learning goals, universal screening should be conducted up to three times a year. The screening data will show the effects of the core math instruction as well as which students need additional help.

References

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

National Research Council. (2001). Adding it up: Helping children learn mathematics.  J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study Committee. Center for Education, Division of Behavioral and Social Sciences and Education.  Washington, DC: National Academy Press.

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