Guest blog by Kathy Kellman, executive editor of mathematics at Curriculum Associates
Note: This is part two of a two-part series. Last week, my colleague Adam Berkin wrote the first part in this series, " Quick Guide to the Common Core: Key Common Core Expectations Explained " for the English Language Arts standards.
A lot of people (including some educators) have a lot of anxiety about math: How do we teach it? How do we learn it? How do we remember it all and use it correctly in real life? Math actually is a logical, accessible, and useful subject, but not enough people know or experience that.
All of mathematics is built on a few basic ideas. The Common Core State Standards aim to have all students thoroughly understand these very few basics (such as how the base-10 system works and what it means to multiply). Students are then asked to use their understanding of these basics, along with important habits of mind such as perseverance and reasoning, to become proficient with the full set of math content standards.
Following are some of the key differences between the new standards and many of the current educational standards in place around the country.
Narrower and deeper focus in each grade
The Common Core standards for math were designed to focus instruction on fewer topics each year, allowing more time to be spent on each topic to foster deeper understanding of key concepts and skills. In elementary grades, most of the focus should be in developing number concepts and skills, including algebraic thinking. For example, in kindergarten, the greatest focus is on developing a strong understanding of numbers--number names, counting, and comparing, including comparing that involves adding (more) and subtracting (less). This deep understanding of number concepts is extended further in first grade, such that students fully understand the quantities represented by numbers and the relationships represented by operations such as addition and subtraction. Allowing students time to develop a deep mastery of these fundamental math concepts will make them much more prepared for learning more complex concepts as they progress in mathematics.
Coherent connections and consistent progressions
The goal behind the demand for coherence is to make math make sense. It requires that learning objectives and instruction be organized to take advantage of the natural connections among standards. Historically, traditional textbooks and teaching methods have often taught mathematical skills in isolation--addition facts one week, subtraction facts another week. But in fact, these and many more topics are interrelated. For example, understanding subtraction is closely tied to understanding addition. The power and beauty of these connections is how it reduces and simplifies the amount of math to be learned; for example, if you know the addition fact 2 +3 = 5, then you already know the subtraction fact 5 - 3 = 2.
Instruction should be consistent with the progressions in the standards and tie new understandings to prior knowledge. Rather than presenting each math topic in isolation, consistent cross-referencing of skills across grades and topics allows students to recognize math as a unified set of related concepts. This empowers students to access skills they already know and reason their way out of confusion about a new topic, skill, or fact. For example, students who know that 8 + 8 = 16 can reason that if 16 - 8 = 8, then 15 - 8 must be 1 less than 8, or that 160 - 80 = 80. If students learn basic principles (e.g., number sense, addition, subtraction), they will be able to build full understanding of larger and more complex topics.
A coherent curriculum makes connections between topics in different domains and across grades as well. Lessons developed from the "Measurement and Data" domain serve students best when they support the skills embedded in the "Operations and Algebraic Thinking" and the "Number and Operations" domains for a given grade level. For example, in grade three, a student will be able to learn "fractions of an inch" because they understand what a fraction is, while measuring fractions of an inch in turn gives meaning and context to support and reinforce learning fractions. Finally, a coherent curriculum applied to skills and concepts across grades means that, in grade four, this same student will be ready to apply what they learned about fractions in grade three to the concept and practice of adding fractions and adding fractional measurements.
Balance between conceptual understanding, procedural fluency, and application
The Common Core standards demand a balance between conceptual understanding (mathematical principles and relationships, or "why"), procedural fluency (skill in carrying out procedures, or "how"), and application (solving real-world problems, or "when"). Each of the three should get individual focus, but students also benefit when they are taught in conjunction with one another. The new standards place equal importance on all three components. Building on the idea of coherent connections and consistent progressions outlined above, the balance of the three gives students a more complete picture of the math they are responsible for learning. Taken together, the three-pronged approach allows students to understand how to carry out a procedure, why it works, and when it applies, particularly in real-world situations.
The word "rigor" is used quite often in conjunction with the Common Core and, in math, applies not just to single standards but to collections of standards. Taken together, the math standards demand students acquire a balance of conceptual understanding, procedural fluency, and problem-solving skills. In a vein similar to increased text complexity on the ELA standards, math standards in the Common Core are asking students to engage in more complex, higher-order thinking. They require students to gain in-depth mastery of skills, to demonstrate mastery by solving non-routine and multi-step problems, and to fully explain and justify their solutions.
In addition to the content standards, the Common Core outlines eight standards for mathematical practice. These mathematical practices are the habits of mind that mathematicians and successful math students use. Similar to the NCTM (National Council of Teachers of Mathematics) process standards, these are not stand-alone standards, but standards that support students in their efforts toward mastery of the content standards. The Common Core takes these process standards even further by specifying that students should develop habits of noticing and making use of the recurring structures and patterns in math. For example, paying attention to and using the power of the base ten system allows students to make connections, solve problems, and be confident in developing new skills.
Success in math hinges on a rich learning environment that provides ample opportunities for teachers to model the practices during instruction, and to point out when students are or could be implementing them. For example, classrooms and curricula in which problem-solving, reasoning, and discussion are prevalent naturally support perseverance and analytical thinking, as well as provide motivation for modeling, precision, and attention to structure and regularity in math--all of which are part of the standards for mathematical practice.
A key element of mathematical practice or "habits of mind" is mathematical reasoning. The Common Core standards expect mathematical reasoning to play a major role in students' learning because through reasoning students can develop ownership, confidence, and mastery. Students should be asked to reason and to explain their reasoning, and instruction should consistently emphasize the role of reasoning in the field of math. For example, a student might have a solid understanding of how to add numbers with two digits. When it's time for that student to learn three- and four-digit addition, they can reason, based on their knowledge of two-digit numbers, how to apply the same base ten principles to add these larger numbers.
Curriculum should incorporate mathematical practices, instruction should be heavily weighted toward reasoning, and math should be taught as a subject that makes sense. One mathematical idea stems from another in a simple, logical way. The role of the teacher is not to dispense rules, but to guide and empower students to build on prior knowledge, extend their understanding to solve new problems and apply broader mathematical principles, and embrace the power of math as a tool for success in life.
As evidenced by the highlights above, the standards "were not so much assembled out of topics as woven out of progressions. Maintaining these progressions in the implementation of the standards will be important for helping all students learn mathematics at a higher level" (Publishers' Criteria for the CCSS). Many educators teach in a way that is aligned to the construct of the standards--and did so even before the standards existed. However, most materials developed by publishers have not supported these efforts enough. We hope to change that.
As Adam mentioned in his article last week, many of us at Curriculum Associates are former teachers, and our team is constantly striving to do what's right--and what's best--for students and teachers, and to make successful learning a better and more positive experience for everyone involved. That is why we continue to work so hard to understand both the major shifts and nuances to create materials that support students and teachers at every step in the process. We believe that the cohesive nature of the standards combined with the level of detail provided to help educators implement them with flexibility and fidelity has the potential to raise the bar for all of us--and enable teachers to educate all of our children better and at a higher level than we ever have before.
For more information on the Common Core State Standards, please visit: http://www.corestandards.org/.
Kathy Kellman is Executive Editor of Mathematics at Curriculum Associates and has an extensive background in publishing and math education. Previously, she contributed toward development of an innovative Singapore Math program for a leading publisher. She has been a principal and curriculum coordinator for a middle school for girls from low-income families, and taught elementary, middle, and high school math and science for nearly a decade.