What is high-quality mathematics instruction and why is it important?

Page 3: Standards-Based Mathematics Curriculum

Recall that one key element of high-quality mathematics instruction is a standards-based curriculum, which includes the concepts and skills believed to be important for students to learn.  Although past mathematics curricula focused on teaching rules and procedures, as well as on building computational fluency, educators now understand that students require conceptual understanding to correctly apply what they have learned in real-world situations and settings. As such, teachers must provide instruction on both mathematics procedures and mathematics concepts and make the links between the two. Additionally, teachers must adhere to state mathematical standards throughout the instructional process. While they vary by state, these standards address standards for mathematical content and standards for mathematical practice. Let’s take a closer look at each.

Standards for Mathematical Content

Standards for mathematical content describe what knowledge and skills students should learn at each grade level. The NCTM has identified five content standards students should know. These standards are threaded across all grade levels and include:

1. Number and Operations: Involves understanding numbers, their properties, and how they are used in computations.
2. Algebra: Involves understanding concepts like patterns, functions, equations, and the language of mathematics for describing relationships.
3. Geometry: Involves understanding the properties and relationships of shapes and their movements in space.
4. Measurement: Involves understanding measurable attributes of objects, systems of measurement, and the application of measurement techniques and formulas.
5. Data Analysis and Probability: Involves formulating questions and collecting, organizing, and analyzing data to make related inferences and predictions.

Kindergarten – Eighth Grade

Many state standards break down the NCTM content standards into specific mathematical concepts. The table below highlights these concepts and the grade level(s) where they are addressed.

	K	1st	2nd	3rd	4th	5th	6th	7th	8th
Numbers and Operations
Counting & Cardinality	✓
Number & Operations in Base Ten	✓	✓	✓	✓	✓	✓
Number & Operations – Fractions				✓	✓	✓
The Number System							✓	✓	✓
Algebra
Operations & Algebraic Thinking	✓	✓	✓	✓	✓	✓
Ratios & Proportional Relationships							✓	✓
Expressions & Equations							✓	✓	✓
Functions									✓
Measurement & Data
Measurement & Data	✓	✓	✓	✓	✓	✓
Geometry
Geometry	✓	✓	✓	✓	✓	✓	✓	✓	✓
Statistics & Probability
Statistics & Probability							✓	✓	✓

Adapted from the National Governors Association Center for Best Practices & Council of Chief State School Officers (2010), New York State Education Department (2019), and Tennessee Department of Education (2021).

High School

Unlike the K–8 standards that are organized by grade level, the high school standards are separated into categories that often cut across all concepts of mathematics being taught:

• Number and Quantity (Numbers and Operations)
• Algebra
• Functions (Algebra)
• Modeling (Algebra)
• Geometry
• Statistics and Probability

Standards for Mathematical Practice

Standards for mathematical practice describes how students will engage with the content they need to acquire at each grade level. The eight practices (listed in the table below) are applicable to all grade levels, though students might apply them differently based on their age or level of development.

Standards for Mathematical Practice

Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and critique the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning

Adapted from the National Governors Association Center for Best Practices & Council of Chief State School Officers (2010), New York State Education Department (2019), and Tennessee Department of Education (2021).

Curricular Materials

To teach the required standards-based curriculum, educators should use high-quality instructional materials (HQIM). These curricular materials include textbooks, supplemental materials, and activities (e.g., discussions, partner work, computer-based programs). Such materials:

• Align to the standards
• Are evidence based
• Are scaffolded for all students
• Include frequent assessment of student understanding

Educators should keep in mind that textbook developers are strongly motivated to claim that their products align with current mathematical standards or a specific state’s standards. Therefore, they should avoid taking such pronouncements at face value and instead personally evaluate curricular materials to assess their usefulness for all students, including those with MLD and English learners (ELs). If the curricular materials do not align with state standards or otherwise fail to address the needs of all learners, educators must supplement them. Fortunately, many instructional features have been shown to be effective for all students—particularly for those with MLD, ELs, and those from economically disadvantaged backgrounds. Although these instructional features are sometimes built into curricula or appear in supplemental materials for diverse learners, other times they are missing altogether. When considering instructional materials, in addition to ensuring they align with grade-level standards, educators should ensure that they include most or all of the following instructional features.

Instructional Features	Guiding Questions
Clear objectives	Are the lesson objectives specific enough that student mastery can be easily determined?
Teaching of single skill or concept	Does the lesson focus on a single skill or concept and connect to prior knowledge?
Use of concrete visual representations	Does the lesson promote the use of concrete, hands-on manipulatives to demonstrate concepts and procedures and to aid mathematical problem-solving?
Use of semi-concrete visual representations	Does the lesson use two-dimensional or virtual representations that illustrate concepts and procedures and that can be used as a tool for problem-solving?
Concurrent use of concrete and semi-concrete representations	Does the lesson connect concrete and semi-concrete representations to facilitate mathematical learning?
Instructional approach	Does the lesson incorporate the use of explicit instruction?
Teacher examples	Does the lesson provide sufficient examples that the teacher can use to demonstrate a skill or concept?
Opportunities to practice	Does the lesson include ample practice problems to help students understand the concept or master the skill, and does it include frequent and strategic practice of previously taught material?
Review of prerequisite skills	Does the lesson review the prerequisite skills needed to learn the new skill?
Error correction and corrective feedback	Does the lesson integrate corrective feedback that includes systematic procedures to help students correctly solve problems?
Vocabulary	Does the lesson identify key vocabulary terms and describe how the teacher should introduce, define, and use terms during instruction?
Strategies	Does the lesson include step-by-step strategies (explicitly taught by the teacher) that students can use to solve problems?
Formative assessment	Does the lesson provide ongoing formative assessment for the teacher to assess a student’s learning and mastery to adjust teaching when necessary?
Progress monitoring	Is formal progress monitoring aligned with the curriculum to demonstrate progress throughout the year?

Adapted from Bryant et al. (2008) and Doabler et al. (2012).

In addition to the instructional features that are effective for all students, educators should ensure that their curricular materials address the linguistic needs of ELs. Learn more about instructional features that can address these needs.

For additional information about content discussed on this page, review the following resources. Please note that these resources are not required readings to complete this module. Links to these resources can be found in the Additional Resources tab on the References, Additional Resources, and Credits page. Guidelines for Design of Mathematics Instruction and Materials for ELLs This document provides detailed guidelines specific to ELs that teachers can use to assess curricular materials and design instruction. Common Core State Standards for Mathematics This document provides K–12 mathematics standards that are designed to ensure clear, specific, and coherent curricula for states, including the concepts that students need to understand and the benchmarks they must meet. Executive Summary: Principles and Standards for School Mathematics This document provides an overview of the six principles, as well as the Pre-K–12 curriculum standards, necessary for a high-quality mathematics education.