UCSMP Grades 6-12 and the Mathematical Practices in the CCSS-M
The Common Core State Standards first appeared in 2010, after the seven books of the UCSMP third edition had been written, tested, modified to capitalize on what was learned in the testing, and published by Wright Group/McGraw-Hill. Thus UCSMP materials were available to the drafters of the CCSS. From a number of the major features of both the mathematical practices and the grade level standards for mathematical content, it seems that they were influenced by the work done by UCSMP and others. New features in the CCSS for grades 6-12 that have been mainstays of the UCSMP curriculum include a strong emphasis on algebra and geometry in courses before the first full algebra course, the incorporation of a substantial amount of the study of statistics as part of the standard curriculum, and the use of geometric transformations as a unifying idea in geometry that applies also to the study of algebra and functions.
In other documents we have noted that the UCSMP materials cover all of the standards for mathematical content. In this document, we explain how the eight mathematical practices identified in the CCSS relate to principles and practices that are present in all the UCSMP materials.
The mathematical practices constitute the glue that holds together the content in UCSMP materials.
Mathematical Practice 1: Make sense of problems and persevere in solving them.
Making sense of problems. UCSMP materials have five broad practices to help students make sense of problems:
- careful attention to representations, both concrete and iconic (graphical) – the U of the SPUR approach;
- uses of the mathematics, to put problems in context – the U of the SPUR approach;
- careful language, to help students decipher the meanings of what they read – summarized in a vocabulary list for each chapter;
- activities in which students conjecture, try special cases, and generalize; and guided examples to help students through more challenging content;
- strong attention to ways of checking answers as well as strong attention to alternate ways of approaching the same problem.
- These practices are present in virtually all lessons in all years.
Perseverance. Perseverance requires that students engage in mathematical activities that are of longer duration than typical classroom problems. In every lesson there are one or two problems that are likely to require a good deal of thought before solution. There is also at least one exploration question designed to extend a student’s knowledge of something in the lesson. And in every chapter there is a set of projects that might require several hours work and that can be assigned to all students or viewed as extra credit.
Mathematical Practice 2: Reason abstractly and quantitatively.
This practice is detailed in the CCSS by three phrases: the ability to decontextualize, the ability to contextualize, and quantitative reasoning. These are hallmarks of UCSMP materials.
The first two abilities require that there be contexts with which students can work. In UCSMP materials the contexts are both real-world and abstract; within a chapter each lesson is connected to the lessons around it by the use of similar contexts so that a student becomes comfortable with relating the mathematics to the contexts.
Quantitative reasoning requires that students confront data such as that they might see every day in the real world, a hallmark of UCSMP materials at all levels. The careful use of units, identified as an important aspect of quantitative reasoning, is also a feature of UCSMP materials at all levels. This mathematical practice also requires the careful use of language, another UCSMP feature.
Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.
Construct viable arguments. Students must be taught how to construct arguments. For this, they must see such arguments in their books and classrooms, and the construction of arguments must be an expectation. This is done in a number ways.
- In all examples, what a student is expected to write in response to a question is shown in a special font.
- In guided examples, students see the framework of an argument and have to fill in the details.
- Beginning in the middle school and continuing through all courses, students are expected to be able to justify procedures by invoking broad mathematical principles such as the Distributive Property. This is the reasoning dimension P of the SPUR approach.
- A major emphasis on writing proofs is found in the last chapter of UCSMP Algebra intended for top-performing students and throughout UCSMP Geometry. Proof arguments are then found through UCSMP Advanced Algebra and Functions, Statistics, and Trigonometry. A second major emphasis on proof is in Precalculus and Discrete Mathematics so that a student who has completed all the UCSMP courses will be prepared for proof-oriented calculus classes either later in high school or in college.
Critiquing the reasoning of others. Many of the activities throughout UCSMP materials are designed to be done in groups so that students become accustomed to discussing their thinking with their classmates. But textbooks and ancillary materials can only do so much in this regard. Consequently, in the Teacher’s Editions, the notes accompanying each lesson provide more ideas for teachers to engage students in discussing the mathematics they are doing.
Mathematical Practice 4: Model with mathematics.
Applying mathematics to the real world is a hallmark of UCSMP materials not just because of the interesting applications to be found in virtually all lessons in all courses but also because of the learning progression (unique to UCSMP) that starts with uses of numbers (counts, measures, ratio comparisons, scale values) and models for arithmetic operations in Pre-Transition Mathematics, reviews and extends these models and introduces uses of variables in Transition Mathematics. In Algebra, Advanced Algebra, and Functions, Statistics, and Trigonometry the models become extended to models for linear, quadratic, exponential, and trigonometric functions. The purpose of the overt discussion of models is to provide students with the principles that connect mathematics with the real-world.
In geometry, a parallel approach is used with uses of points (as locations, as ordered pairs, as nodes of networks, and as dots) being the basics from which applications of geometry arise.
The use of real data throughout, some presented in the texts, some gathered by students, provides more opportunities to see how mathematics can be used to understand phenomena and solve problems in the real world. In all the UCSMP courses, this powerful approach is embodied in the U (uses) of the SPUR approach to understanding.
Mathematical Practice 5: Use appropriate tools strategically.
The first three sentences of the CCSS description of this practice embody the UCSMP view on the use of tools. "Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations."
The use of the latest calculator and computer technology is another hallmark of UCSMP materials. The UCSMP curriculum for middle and high school was the first full curriculum in the U.S. to require scientific calculators, the first to incorporate graphing technology, and the first to make use of computer algebra systems. In all these efforts, we have taken into account that, outside of school, technology to do mathematics is ubiquitous and that a person who is not equipped to use the technology wisely is behind the times. Thus we ask students not just to experiment or explore with these technologies, but to use them to do the mathematics required to answer both routine and non-routine questions. Our experience with these and other technologies makes it clear that students have to be taught to use them as part of their everyday mathematics experience and not just as tools to be taken out sporadically. It is with everyday contact that students learn best when to use them and when not, and how to use these tools wisely.
Paper-and-pencil is also a "technology", and today’s students need to learn to use that technology as well, and use it wisely. In general, we expect students to be able to answer many questions "in their head"; questions to be answered mentally are found in every lesson. Students should learn how to answer questions requiring only a modicum of paper and pencil work both with paper and pencil and with electronic technologies, with each checking the other.
Mathematical Practice 6: Attend to precision.
The CCSS description of this mathematical practice details what is meant by "attend to".
- Communicate precisely to others.
- Try to use clear definitions in discussion with others and in their own reasoning.
- Are careful about specifying units of measure and labeling axes.
- Calculate accurate and efficiently. The ideas of these bullets have been mentioned above in connection with other mathematical practices. All of these are fundamental goals throughout all the UCSMP materials.
- Express answers with a degree of precision appropriate for the problem context. It is important for students to know not only when their answer is not precise enough, but when it is too precise. In many real situations (how long will a light bulb last, how much does an elephant weigh, what does a teacher earn in a year), an estimate or an interval is preferred over an exact value. The UCSMP emphasis on data throughout, coupled with the in-depth discussion of probability and statistics in the various courses, forces an attention to precision.
- Examine claims. Activities in all the courses have students testing claims or conjectures. Claims that can be tested using deductive reasoning are also found throughout. Claims that can be tested with statistics are studied in Pre-Transition Mathematics, Algebra, and Functions, Statistics, and Trigonometry. Technology may be employed to show that a claim is false but great care is taken to point out that the truth of a mathematical argument rests on deduction.
Mathematical Practice 7: Look for and make use of structure.
Structure has always been important to the organization of content in the UCSMP courses because students are far more likely to learn and remember what they have learned if they can place the content in a larger structure. This principle holds for all learning, but in mathematics it is particularly special because of the special nature of truth; a statement must either be assumed or be able to be deduced from assumptions in order to be considered true. In the UCSMP middle school courses, students learn that one use of variables is as pattern generalizes and they are asked often to describe patterns using variables. In UCSMP Geometry, students work from the postulates and definitions of Euclidean geometry and see how theorems can be deduced; they practice doing simple proofs. The language of deduction continues as a backdrop in all the later courses; students learn that properties are either postulates, can be deduced from the postulates, or can be deduced from definitions. In Precalculus and Discrete Mathematics, students are introduced to the logic of deduction to provide a further bulwark to mathematical thinking that started many years before with the generalization of arithmetic patterns.
Mathematical Practice 8: Look for and express regularity in repeated reasoning.
This goal is one to which all teachers strive. It is the reason we ask students to practice whatever tasks we wish them to perform with success. UCSMP also works to avoid three dangers associated with excessive practice too early in the learning sequence: the learning of wrong principles with the result that a student is practicing the mistake; the overuse of inductive reasoning to reason from a pattern when the pattern does not generalize; and the forgetting of something as quickly as it was learned.
To avoid the learning of wrong principles, technology is constantly available to show correct answers to arithmetic and algebraic work; this provides instant feedback to alert to wrong answers and to reinforce correct ones.
To avoid the overuse of inductive reasoning, students are asked to justify their work by referring to known mathematical properties.
To avoid quick forgetting, review questions are a part of the question set in each lesson. Within a chapter, these questions reinforce what was learned in the lessons immediately preceding, and at the end of the chapter we strongly recommend covering a large set of review questions covering skills, properties, uses, and representations for that chapter, as our testing indicates that this review substantially increases performance. Ideas from each chapter are then reviewed in succeeding chapters to maximize long-term retention.
Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content.
The CCSS have this to say about these connections.
"The Standards for Mathematical Content are a balanced combination of procedures and understanding. Expectations that begin with the word "understand" are often especially good opportunities to connect the practices to the content."
We at UCSMP have always felt that "understanding mathematics" involves concepts, procedures, and problems working together.
The SPUR approach to understanding ensures that students are continually connecting their skill work (S) with the mathematical principles and properties (P) that underlie our subject, with the uses (U) that are the main reason students are asked to learn a great deal of mathematics and the representations (R) that are so helpful in learning these ideas. The SPUR approach is employed in the UCSMP materials as an organizer of reviews in each chapter, as a way to ensure that tests and quizzes are balanced in their treatment of content, and because we believe that the best mathematics education for a student is one that balances theory and practice, conceptual work and problem-solving, and as it does these things, displays the value and importance of learning mathematics.
The SPUR (S = skills; P = properties, U = uses, R = representations) approach to understanding is UCSMP’s way of ensuring that mathematical concepts and practices are not undermined by meaningless drill and practice.
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