What is high-quality mathematics instruction and why is it important?
Page 2: A Standards-Based Mathematics Curriculum
Among experts, the debate over the most effective method of teaching mathematics is hardly new. In the past, mathematics curricula focused on teaching rules and procedures, as well as on building computational fluency. Though the importance of this procedural knowledge is beyond question, it’s also the case that students require conceptual understanding if they are to correctly apply what they have learned in real-world situations and settings. In other words, teachers must offer instruction on both mathematics concepts and procedures and make the links between the two explicit.
Traditional curricula have likewise been criticized for offering shallow content, failing to connect knowledge and skills, over-emphasizing computational procedures, and making only minimal use of evidence-based practices. To address these shortcomings, educators are urged to implement standards-based curricula. These focus on the conceptual understanding of mathematical principles (the why) and the connection between them as outlined by mathematic standards, rather than focusing heavily on procedural knowledge (the how).
To ensure that their students acquire the skills they will need to be college- and career-ready, many states have adopted increasingly rigorous standards. In the case of mathematics, for most states this has meant the Common Core State Standards for Mathematics (CCSSM), which outline the skills and concepts students are expected to learn at each grade level. More specifically, the CCSSM:
- Build upon the strengths of current state standards
- Are informed by instructional practices used in other top-performing countries, so that all students are prepared to succeed in our global economy and society
- Are evidence-based
- Are aligned with college and work expectations
- Are clear, comprehensible, and consistent
- Embrace rigorous content and require the application of knowledge through higher-order skills
- Encourage the use of real-world problems
For Your Information
As with many areas touched by politics, a great deal of controversy has surrounded the Common Core State Standards; however, the vast majority of the objections raised about them have been based on either misunderstanding or misinformation. The facts are these: The CCSS were developed as part of a state-led initiative—sponsored by the Council of Chief State School Officers (CCSSO) and the National Governors Association (NGA)—to create a shared set of learning standards for schools in the United States. This effort was informed by teachers, administrators, and experts and reflects the best instructional models used both in the United States and internationally.
Diane Pedrotty Bryant discusses the purposes of the CCSSM (time: 2:20).
Diane Pedrotty Bryant, PhD
Project Director, Mathematics Institute for Learning Disabilities and Difficulties
University of Texas at Austin
Transcript: Diane Pedrotty Bryant, PhD
The Common Core State Standards in mathematics are a really good attempt to provide more standardized mathematics standards across states. There’s has been a lot of criticism about the math curriculum in schools that’s been voiced by professional organizations and the National Mathematics Advisory Panel about the mathematics curriculum and what we’re trying to cover in various grade levels. There’s a saying that it’s a mile wide and an inch deep, suggesting that we’re trying to do too much as part of math instruction in classrooms. So I think the standards are really intended to provide a more focused and coherent set of standards around specific mathematical topics and to be sure that these topics are developmentally appropriate, that they follow learning progressions of mathematics topics and also build conceptual understanding as well as procedural knowledge. There’s a real focus on helping students develop more of that conceptual understanding of mathematics that sometimes is missing in instruction.
And I think it’s important to mention that the standards don’t purport to be a mathematics curriculum or a set of specific interventions for teaching mathematics. That’s really up to school district leaders to operationalize the standards according to their own district guidelines and district curricula. The standards are not intended to dictate interventions and curriculum in classrooms but to provide guidance for district leaders to make decisions about what are the critical mathematical ideas that need to be taught in each grade level and what materials do we have in district programs that can help teachers adjust these standards.
The CCSSM are composed of eight Standards for Mathematical Practice that describe how students will engage with mathematical content, as well as the Standards for Mathematical Content, which outline what students should learn at each grade-level.
This graphic illustrates the components that make up the Common Core State Standards for Mathematics (CCSSM). A pair of rectangular boxes represent individual standards and their intended effect on student learning. These are connected by an arrow to a larger rectangle just beneath, which is labeled “Common Core State Standards for Mathematics (CCSSM).”
On the top left is a blue box labeled “Standards for Mathematical Practice: How student will engage.” The box is further illustrated with a compass, a set square, and a ruler. On the top right is a yellow box labeled “Standards for Mathematical Content: What student should learn.” The box is further illustrated with an open math textbook. The blue box and the yellow box connect to one another at their bottoms to form an arrow, where their colors blend to form green, which is the color of the Common Core State Standards for Mathematics (CCSSM) box into which they flow.
Standards for Mathematical Practice
The CCSSM Standards for Mathematical Practice—which are themselves based on standards developed by the National Council of Teachers of Mathematics (NCTM) and the National Research Council—describe how students will engage with the content they need to acquire at each grade level. These eight practices (which you will find 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.
National Council of Teachers of Mathematics (NCTM)
The world’s largest organization of mathematics education professionals. In 1989, the NCTM developed the Principles and Standards for School Mathematics. Revised in 2000, these standards served to guide mathematics instruction in the United States by outlining the content and skills that compose high-quality mathematics curriculum for grades K–12.
National Research Council (NRC)
An organization that convened a team of experts to explore the reasons for poor mathematics performance among students in pre-K through the 8th grade in the United States. The resulting report, Adding It Up: Helping Children Learn Mathematics, offers recommendations about how to improve instruction, instructional materials, and teacher education in mathematics.
|CCSSM Standards for Mathematical Practice|
|MP1: Make sense of problems and persevere in solving them.
MP2: Reason abstractly and quantitatively.
MP3: Construct viable arguments and critique the reasoning of others.
MP4: Model with mathematics.
MP5: Use appropriate tools strategically.
MP6: Attend to precision.
MP7: Look for and make use of structure.
MP8: Look for and express regularity in repeated reasoning.
Standards for Mathematical Content
The CCSSM Standards for Mathematical Content describe the knowledge and skills that students should acquire. These are divided into domains that vary by grade level.
Kindergarten – 8th Grade
The standards for kindergarten through grade eight are categorized into 11 domains. The table below highlights these domains and the grade level(s) in which each of them is addressed.
|Counting & Cardinality|
|Operations & Algebraic Thinking|
|Number & Operations in Base Ten|
|Measurement & Data|
|Number & Operations – Fractions|
|Ratios & Proportional Relationships|
|The Number System|
|Expressions & Equations|
|Statistics & Probability|
Adapted from Common Core State Standards Initiative
The purpose of the CCSSM is to prepare students for college and the workforce by helping them to understand thoroughly mathematical concepts and procedures, as well to develop the ability to transfer that knowledge to novel situations. This ability is called generalization. Unlike the K–8 standards that are organized by grade level, the high school standards are separated into conceptual categories:
- Number and Quantity
- Statistics and Probability
For Your Information
The CCSSM Website is a good resource for learning more about these standards. The links below might be of particular interest.
- Standards for Mathematical Practice
- Standards for Mathematical Content
- Standards in Your State (Click this link to determine whether your state has adopted the CCSSM.)
Did You Know?
The U.S. Department of Education has identified a number of exemplary standards-based mathematics programs. Click the link to learn more about them.
To teach the required standards-based curriculum, teachers use curricular materials—that is, any of a variety of resources, items, or tools designed to engage students in the learning process. These curricular materials include textbooks, supplemental materials, and activities. Before they are used in the classroom, however, teachers (or school teams) must first assess not only new materials but also existing ones to make sure they align with their state’s standards and address the needs of diverse learners. Fortunately, a number of instructional features have been proven effective for students with MLD and might prove beneficial for other students who struggle with mathematics as well.
|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?|
|Use of manipulatives||Does the lesson promote the use of concrete objects to demonstrate concepts and procedures and to aid mathematical problem solving?|
|Instructional approach||Does the lesson incorporate the use of evidence-based instructional strategies such as explicit, systematic 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?|
|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 and define the terms?|
|Strategies||Does the lesson include step-by-step cognitive strategies (explicitly taught by the teacher) that students can use to solve problems?|
|Assessment||Does the lesson provide opportunities for the teacher to individually assess a student’s learning and mastery of the skill or concept?|
Adapted from Bryant, Bryant, Kethley, Kim, Pool, & Seo (2008).
Just as they ensure that their curricular materials address the needs of students with MLD, teachers must do the same for students who are ELLs. The document below provides detailed guidelines specific to ELLs that teachers can use to assess curricular materials and design instruction.
Guidelines for Design of Mathematics Instruction and Materials for ELLs
An important factor to keep in mind is that textbook developers are strongly motivated to claim that their products align with current mathematical standards or a specific state’s standards. Teachers should take care not to accept such pronouncements at face value and instead personally evaluate curricular materials to assess their value. In the event that the curricular materials do not align with state standards or otherwise fail to address the needs of diverse learners, teachers must supplement them. Listen as Kim Paulsen provides more information about supplementing the curriculum (time: 1:38).
Kim Paulsen, EdD
Associate Professor, Special Education
Transcript: Kim Paulsen, EdD
One thing that you have to do is look at curriculum that your district has selected, looking at how much time is devoted to math at your grade level, and then what can you do with that curriculum? Will that cover everything? Most of the time, the answer to that is no. And it’s not that the material in the curriculum isn’t designed well, but that it’s not enough time and there’s usually not enough practice. For example, in teaching time in the first grade you might have one or two pages in your math textbook and students are expected to know it, and that’s not going to happen. And so having supplemental materials that will help with that—and that could be things from your district-wide publishing company that you use, or you can find those materials on Websites or by purchasing different technology games that actually help students. We need to look at kids that are having difficulty or that didn’t understand one concept—what is the reason for it?—and then find materials. And then think about the students who have already mastered that concept, and what do you have them do while you’re working with the students who still maybe need a little more instruction? And we want to make sure that when we’re working with the students who have already understood the concept that we aren’t just giving them busy work. We want to have materials that challenge them and that really focus more on the higher-level thinking skills.
For Your Information
As they work to meet the needs of their struggling students, teachers must not overlook those who excel in mathematics. Curricular materials that include enrichment activities will allow these students to delve deeper into topic areas and explore more sophisticated ways to apply mathematical processes.