It is the significance of detail wherein the truth lies. Nadine Gordimer
Mathematics is a broad, multi-faceted, and multidimensional subject; it is about big ideas, deep concepts, elegant procedures, numerous and diverse applications. To think mathematically affords a powerful means to understand and control one’s social and physical reality. An important aspect of learning mathematics is to become aware of the fact that mathematics is indeed a useful tool for action and understanding. Yet, despite some 12 or so years of compulsory mathematical education, most children, even in the developed world, leave school with only a limited access to mathematical ideas and their vast applications. The Common Core State Standards in Mathematics (CCSSM) are an attempt to bring the big ideas and their elegance to school mathematics curriculum.
Developed in 2009 through a collective effort by educators and others across the country, the Common Core State Standards are also meant to provide some answers to our intractable school problems: 33% of all students drop out of school in the United States. Only 50% of Latino, African American, and Native American students in the United States complete high school. According to a recent Gates Foundation–funded study, 81% of those who drop out of school claim that “opportunities for real world learning” would have improved their chances of staying in school. 69% were “not inspired to work hard” and 47% said “classes were not interesting.” Significant to these findings is also the fact that only 35% of those interviewed claimed that they left because they were “failing in school.”
To solve these problems, we need to engage students in learning: set up activities and classroom where they are encouraged to ask good questions, map out solution pathways to problems, reason about complex solutions, interpret solutions, set up models and communicate in different forms. We need students who can do real mathematics, not just calculate quickly in math. Doing real mathematics is about inquiring, communicating, making connections, and representing ideas in multiple forms. Doing real mathematics is problem solving, modeling, thinking, and reasoning, as these are the mathematical abilities for the workplace and a technology rich world. This broad, multidimensional mathematics is the math that engages many learners and puts them on a pathway to lifelong success. All of these ways are encouraged by the CCSSM.
The CCSSM’s goal is to achieve a higher level of mathematics competence for American students. For each grade level, K-8 and High School, the CCSSM list the most important mathematical concepts and “what students should understand and be able to do” with them. Unfortunately, there is little guidance for implementing the CCSSM although they are accompanied by suggestions for instructional practices—the Standards of Mathematics Practices (SMP).
Informed by best practices in national and international classrooms, the SMP describe expertise and mindsets that mathematics educators should practice and seek to develop in their students in order to implement the CCSSM. SMP suggestions for implementation of CCSSM are based on National Council of Teachers of Mathematics process standards of problem solving, reasoning and proof, communication, representation, and connections and the mathematical proficiency expectations in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out appropriate procedures flexibly, accurately, and efficiently), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).
The main difference between CCSSM and previous state standards and frameworks for mathematics teaching and children’s achievement in mathematics is that the CCSSM are mathematically sound—they have focus, rigor, and coherence. They seek mastery of the material taught, not just exposure. They are aimed at helping students develop not just content mastery but also mathematical ways of thinking and the ability to apply what they learn. The CCSSM are focused and emphasize depth over breath. Moreover, as standards that set learning targets for children, they are not a curriculum. Hence, they do not mandate how teachers assist children in meeting the targets. Instead, they serve as the foundation for school curricula, mathematics textbooks, teacher preparation, lessons and assignments, student engagement and attitude, and assessment of learning.
The CCSSM provide a “staircase” of content with increasing complexity with the goal that all children become college and career ready. As such, they offer a clear design, common central goals, common language, and common high standards. Cross-curricular teaching that emphasizes problem solving, persistence, abstract reasoning, and the ability to construct arguments and critique reasoning is at the core of these standards. And this is without sacrificing the beauty and rigor of mathematics that we value.
Using the SMP, states and school systems must translate the CCSSM into well-defined curricula and executable lesson plans. When teachers understand the intent and content of these standards, align their curriculum with CCSSM expectations, and convert them into daily lesson plans coupled with appropriate progress monitoring, then children’s mathematics achievement will rise.
The CCSSM call for examining how teachers teach mathematics and how and what students at each grade level learn, know and communicate, not just what is covered. CCSSM content expected to be mastered by students is demanding. There is, therefore, an important distinction between what we as educators need to know vs. what our students need to be taught. To deliver CCSSM effectively, we need to be more than just a page, a lesson, or a concept ahead of students. We need to know much more than what we want our students to know. In addition, we need to select instructional materials wisely because they can have as great an effect on student test scores as teacher knowledge.
The focus of mathematics content in the CCSSM in the elementary grades is on mastering key arithmetic concepts and procedures with deeper understanding so that students are prepared for more demanding and meaningful and comprehensive mathematics during the middle and high school years and beyond.
Such a focus means mastering certain non-negotiable skills at each grade level with conceptual understanding and fluency. Much of what we teach children, at present, during their first decade of math education relies on students’ memorization of rules, tricks, and facts. We reward correct answers, but we do not encourage students to think independently about what these rules, tricks, and facts might mean in the bigger mathematical picture.
Tricks such as telling children to think of a greater-than sign as Pac-Man or to cross-multiply when dividing fractions, or invert and multiply for dividing fractions to help children get the right answers to difficult problems have long been a staple of math education. In contrast, the intent of CCSSM and SMP is for children to know strategies, not tricks. Methods based on shortcuts and tricks are not real mathematics. If we cannot give a child the reasoning for the use of a method, procedure, or trick, then we are not teaching mathematics and the child is not being prepared for problem solving in higher grades or the work place. It is therefore never sensible to have students memorize first and understand later; this approach leaves students unprepared when they move from elementary mathematics to complex problem solving.
At any grade level, the non-negotiable skills are the basis of all other concepts, procedures and skills at that grade level. If the non-negotiable skills are mastered, then other skills and concepts are easily learned. For example, in CCSSM, the concepts of fractions are introduced in earlier grades (e.g., adding and subtracting fractions in the third and fourth grades with same denominators), but by fifth grade, children have mastered the operations on fractions with deeper understanding. Thus, fractions—understanding and mastering of the concept and procedures, is a non-negotiable skill in the fifth grade. However, the mastery of fractions is dependent on (a) mastery of multiplication facts, (b) divisibility tests, (c) prime factorization, and (d) short-division that are learned by the end of fourth grade.
Multiplication is introduced in the second grade as repeated addition, equal groups of objects, and arrays. By the end of third grade, however, children should have mastered the concept of multiplication as repeated addition, groups of, arrays, and the area of a rectangle so that they understand the distributive property of multiplication over addition and subtraction, have automatized multiplication facts, and are prepared to multiply fractions in fifth grade using the area model of multiplication and then apply multiplication of fractions to master the operations of addition and subtraction on fractions with efficiency and understanding. Using the same principles, they should have automatized division facts by the end of fourth grade and division of fractions by sixth grade.
With the current focus on “covering” and “spiraling” many children lack true mastery in concepts, procedures, and skills. They move with “shallow” mastery from grade to grade and with “holes” in their conceptual and skill sets. Because of this lack of mastery teachers presently devote a great deal of time on preview, pre-testing, and review of previous material and do not teach the grade level content—language, concepts, and procedures, with a level of mastery needed to progress in higher mathematics. In such situations, they hurriedly cover the material, and some students never see meaningful mathematics. With a focus on mastery of non-negotiable skills, teachers can pay attention to children’s errors or lack of mastery and provide appropriate remedial instruction in proper time.
Our classrooms have become so diverse in the levels of mastery of content and preparation for mathematics that every teacher has to make several preparations to present the material to the whole class. She ends up teaching the class in several smaller groups, giving limited instructional time to each group while expecting to achieve a year’s growth in return for a fraction of a year in instruction for each child (and therefore the whole class). Each student gets limited attention in the allotted time for instruction; as a result, the teacher finds meeting individual academic needs to be very difficult.
These multiple preparations for the same class with only “shallow” coverage as a focus rather than mastery send many students to the next grade with holes in their preparation and limited achievement and mastery. Even the best strategies of differentiation, in such situations, are inadequate to make up the gaps. On the other hand, breaking each grade into sections according to ability deprives some children the opportunity of novel thinking strategies and others of communication and collaboration. Most ability groupings in American schools are based on students’ computational facility. As a result, ability grouping before sixth grade sends a message that mathematics is just a collection of computational tasks. Recognizing the need for some differentiated instruction, CCSSM does make provision for accelerated mathematics programs at and after sixth grades for students who have conceptual understanding, computational fluency, and adequate cognitive and mathematical thinking that they can handle higher order of thinking and problem solving. These students are the future of the country. They are our future mathematicians, natural and social scientists.
Proper implementation of CCSSM will solve many instructional and achievement problems in our classrooms—particularly the diverse levels of students’ mathematics mastery. SMP and CCSSM are steps in the right direction.
This is the first in a series of blogs examining SMP and CCSSM in American schools.