# Essential Elements | Mathematics

## Introduction

## Standards for Mathematical Practice

Iowa Core Mathematics includes recommendations for curriculum, instruction, and assessment, as well as standards for mathematical content and mathematical practices. In particular, Iowa Core Mathematics is built upon:

- Iowa Core’s Characteristics of Effective Instruction
- Teaching for Understanding
- Student–Centered Classrooms
- Teaching for Student Differences
- Rigor and Relevance
- Assessment for Learning

- Common Core State Standards for Mathematics
- Standards for Mathematical Practice
- Standards for Mathematical Content

To faithfully implement Iowa Core Mathematics requires teaching with rich mathematical tasks.

**Rich Mathematical Tasks**

As stated in Common Core Mathematics, “Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.” (p. 6). By teaching through rich mathematical tasks, students develop deep conceptual understanding and skill proficiency. Rich mathematical tasks involve both:

- Teaching through Problem Solving with Problem–Based Instructional Tasks
- Using Distributed Practice that is Meaningful and Purposeful

**Problem–Based Instructional Tasks**

Problem–based instructional tasks are at the heart of teaching for understanding. A world–class mathematics curriculum should be built around rich instructional tasks focused on important mathematics.

Problem–based instructional tasks:

- Help students develop a deep understanding of important mathematics
- Emphasize connections, across mathematical content areas, to other disciplines, and especially to the real world
- Are accessible yet challenging to all
- Can be solved in several ways
- Encourage student engagement and communication
- Encourage the use of connected multiple representations
- Encourage appropriate use of intellectual, physical, and technological tools

**Distributed Practice that is Meaningful and Purposeful **

Practice is essential to learn mathematics. However, to be effective in improving student achievement, practice must be meaningful, purposeful, and distributed. Meaningful Purposeful Distributed Practice:

- Meaningful: Builds on and extends understanding
- Purposeful: Links to curriculum goals and targets an identified need based on multiple data sources
- Distributed: Consists of short periods of systematic practice distributed over a long period of time

**Toward greater focus and coherence **

*Mathematics experiences in early childhood settings should concentrate on (1) number (which includes whole number, operations, and relations) and (2) geometry, spatial relations, and measurement, with more mathematics learning time devoted to number than to other topics. Mathematical process goals should be integrated in these content areas.*

– Mathematics Learning in Early Childhood, National Research Council, 2009

*The composite standards [of Hong Kong, Korea and Singapore] have a number of features that can inform an international benchmarking process for the development of K–6 mathematics standards in the U.S. First, the composite standards concentrate the early learning of mathematics on the number, measurement, and geometry strands with less emphasis on data analysis and little exposure to algebra. The Hong Kong standards for grades 1–3 devote approximately half the targeted time to numbers and almost all the time remaining to geometry and measurement.*

– Ginsburg, Leinwand and Decker, 2009

*Because the mathematics concepts in [U.S.] textbooks are often weak, the presentation becomes more mechanical than is ideal. We looked at both traditional and non–traditional textbooks used in the US and found this conceptual weakness in both.*

– Ginsburg, et al., 2005

*There are many ways to organize curricula. The challenge, now rarely met, is to avoid those that distort mathematics and turn off students.*

– Steen., 2007

For over a decade, research studies of mathematics education in high–performing countries have pointed to the conclusion that the mathematics curriculum in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country. To deliver on the promise of common standards, the standards must address the problem of a curriculum that is "a mile wide and an inch deep." These Standards are a substantial answer to that challenge.

It is important to recognize that “fewer standards” are no substitute for focused standards. Achieving "fewer standards" would be easy to do by resorting to broad, general statements. Instead, these Standards aim for clarity and specificity.

Assessing the coherence of a set of standards is more difficult than assessing their focus. William Schmidt and Richard Houang (2002) have said that content standards and curricula are coherent if they are:

*articulated over time as a sequence of topics and performances that are logical and reflect, where appropriate, the sequential or hierarchical nature of the disciplinary content from which the subject matter derives. That is, what and how students are taught should reflect not only the topics that fall within a certain academic discipline, but also the key ideas that determine how knowledge is organized and generated within that discipline. This implies that to be coherent, a set of content standards must evolve from particulars (e.g., the meaning and operations of whole numbers, including simple math facts and routine computational procedures associated with whole numbers and fractions) to deeper structures inherent in the discipline. These deeper structures then serve as a means for connecting the particulars (such as an understanding of the rational number system and its properties). (emphasis added)*

These Standards endeavor to follow such a design, not only by stressing conceptual understanding of key ideas, but also by continually returning to organizing principles such as place value or the properties of operations to structure those ideas.

In addition, the "sequence of topics and performances" that is outlined in a body of mathematics standards must also respect what is known about how students learn. As Confrey (2007) points out, developing "sequenced obstacles and challenges for student...absent the insights about meaning that derive from careful study of learning, would be unfortunate and unwise." In recognition of this, the development of these Standards began with research–based learning progressions detailing what is known today about how students’ mathematical knowledge, skill, and understanding develop over time.

**Understanding mathematics**

These Standards define what students should understand and be able to do in their study of mathematics. Asking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from. The student who can explain the rule understands the mathematics, and may have a better chance to succeed at a less familiar task such as expanding (a + b + c)(x + y). Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.

The Standards set grade–specific standards but do not define the intervention methods or materials necessary to support students who are well below or well above grade–level expectations. It is also beyond the scope of the Standards to define the full range of supports appropriate for English language learners and for students with special needs. At the same time, all students must have the opportunity to learn and meet the same high standards if they are to access the knowledge and skills necessary in their post–school lives. The Standards should be read as allowing for the widest possible range of students to participate fully from the outset, along with appropriate accommodations to ensure maximum participation of students with special education needs. For example, for students with disabilities reading should allow for use of Braille, screen reader technology, or other assistive devices, while writing should include the use of a scribe, computer, or speech–to–text technology. In a similar vein, speaking and listening should be interpreted broadly to include sign language. No set of grade–specific standards can fully reflect the great variety in abilities, needs, learning rates, and achievement levels of students in any given classroom. However, the Standards do provide clear signposts along the way to the goal of college and career readiness for all students.

The Standards begin with eight Standards for Mathematical Practice.

## How to read the grade level standards

**Standards** define what students should understand and be able to do.

**Clusters** are groups of related standards. Note that standards from different clusters may sometimes be closely related, because mathematics is a connected subject.

**Domains** are larger groups of related standards. Standards from different domains may sometimes be closely related.

**Taxonomy Codes** are unique, alpha-numeric identifiers for each standard.

**Depth-of-Knowledge Codes** are cognitive complexity codes based on Webb’s Depth-of-Knowledge that are assigned to each standard.

These Standards do not dictate curriculum or teaching methods. For example, just because topic A appears before topic B in the standards for a given grade, it does not necessarily mean that topic A must be taught before topic B. A teacher might prefer to teach topic B before topic A, or might choose to highlight connections by teaching topic A and topic B at the same time. Or, a teacher might prefer to teach a topic of his or her own choosing that leads, as a byproduct, to students reaching the standards for topics A and B.

What students can learn at any particular grade level depends upon what they have learned before. Ideally then, each standard in this document might have been phrased in the form, "Students who already know ... should next come to learn ...." But at present this approach is unrealistic—not least because existing education research cannot specify all such learning pathways. Of necessity therefore, grade placements for specific topics have been made on the basis of state and international comparisons and the collective experience and collective professional judgment of educators, researchers and mathematicians. One promise of common state standards is that over time they will allow research on learning progressions to inform and improve the design of standards to a much greater extent than is possible today. Learning opportunities will continue to vary across schools and school systems, and educators should make every effort to meet the needs of individual students based on their current understanding.

These Standards are not intended to be new names for old ways of doing business. They are a call to take the next step. It is time for states to work together to build on lessons learned from two decades of standards based reforms. It is time to recognize that standards are not just promises to our children, but promises we intend to keep.