K-12 Mathematics Common Core State Standards Jaime Aquino, Ph.D. General Manager North Region
Objectives • Develop a understanding of the Common Core Sate Standards in Mathematics by relating their implementation to the past, current and future work of Networks. • Identify the implications of the CCSS Math Standards to instruction, assessment, leadership and professional development.
Why do we study mathematics in school? • Because it’s hard and we have to learn hard things at school. We can learn easy stuff at home like manners. Carrine, K • Because it always comes after reading. Roger, 1 • Because all the calculators might run out of batteries or something. Thomas, 1 • Because it’s important. It’s the law from President Bush and it says so in the Bible on the first page. Jolene, 2 • Because you can drown if you don’t. Amy, K
Why do we study mathematics in school? • Because what would you do with your check from work when you grow up? Brad, 1 • Because you have to count if you want to be an astronaut. Like 10…9…8…blast off. Michael, 1 • Because you could never find the right page. Mary, 1 • Because when you grow up you couldn’t tell if you are rich or not. Raji, 2 • Because my teacher could get sued if we don’t. That’s what she said. Any subject we don’t know – Wham! She gets sued and she’s already poor. Corky, 3
What are standards? • Standards define what students should understand and be able to do. • Standards must be a promise to students of the mathematics they can take with them. • We haven’t kept our old promise and now we make a new one. • What difference will it make?
Lessons Learned After two decades of standards based accountability: • Too many standards • Lack of student motivation • “Cover” at “pace” is a failure • Tells teachers to ignore students • Turn the page regardless • Shrug your shoulders and do what “they” say • Mathematics is not a list of topics to cover • Singapore: “Teach less, learn more”
Lessons Learned • TIMSS: math performance in the US is being compromised by a lack of focus and coherence in the “mile wide, inch deep” curriculum • Hong Kong students outscore U.S. students on the grade 4 TIMSS, even though Hong Kong only teaches about half of the tested topics. U.S. covers over 80% of the tested topics. • High-performing countries spend more time on mathematically central concepts: greater depth and coherence.
Answer Getting vs. Learning Mathematics United States How can I teach my kids to get the answer to this problem? Use mathematics they already know. Easy, reliable, works with bottom half, good for classroom management. Japan How can I use this problem to teach mathematics they don’t already know?
Math Standards • Mathematical Performance: what kids should be able to do • Mathematical Understanding: standards for what kids need to understand • Mathematical Practices: behaviors students need to exhibit in mathematics
Performance • Performance: what kids should be able to do • multiply and divide within 100 • 3rd grade sample
Understanding • Understanding: what kids should understand about mathematics • 3rd grade sample • Understand properties of multiplication and the relationship between multiplication and division. • Table Talk: • Why is that our kids do not perform as well as students in other countries do?
2.15 + 3.1 215 + 31
24 x 5 = 120 4 x 1/2 = 2
Impact of Misconceptions We use ideas we already have (BLUE DOTS) to construct a new idea (RED DOT) John Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally, 2004, page 23.
Standards for Mathematical Practice • Make sense of problems and persevere in solving them. • Reason abstractly and quantitatively. • Construct viable arguments and critique the reasoning of others. • Model with mathematics. • Use appropriate tools strategically. • Attend to precision. • Look for and make use of structure. • Look for and express regularity in repeated reasoning.
There are 125 sheep and 5 dogs in a flock. How old is the shepherd?
A Student’s Response There are 125 sheep and 5 dogs in a flock. How old is the shepherd? 125 x 5 = 625 extremely big 125 + 5 = 130 too big 125 - 5 = 120 still big 125 5 = 25 That works!
Math Standards • Mathematical Performance: what kids should be able to do • Mathematical Understanding: standards for what kids need to understand • Mathematical Practices: varieties of expertise that math educators should seek to develop in their students.
Take the number apart? • Tina, Emma, and Jen discuss this expression: 6×5 1/3 Tina: I know a way to multiply with a mixed number that is different from the one we learned in class. I call my way “take the number apart.” I’ll show you. First, I multiply the 5 by the 6 and get 30. Then I multiply the 1/3 by the 6 and get 2. Finally, I add the 30 and the 2 to get my answer, which is 32.
Take the number apart? Tina: It works whenever I have to multiply a mixed number by a whole number. Emma: Sorry Tina, but that answer is wrong! Jen: No, Tina’s answer is right for this one problem, but “take the number apart” doesn’t work for other fraction problems. Which of the three girls do you think is right? Justify your answer mathematically?
Table Talk • What are your reactions to the sample assessment item? • How does the sample assessment item compare to tasks being assigned in your school? • How does the sample assessment item assesses the three types of standards (performance, understanding and practices)? • What are the implications for mathematics teaching and learning?
Overview of K-8 Mathematics Standards • The K-5 standards provide students with a solid foundation in whole numbers, addition, subtraction, multiplication, division, fractions and decimals • The 6-8 standards describe robust learning in geometry, algebra, and probability and statistics • Modeled after the focus of standards from high-performing nations, the standards for grades 7 and 8 include significant algebra and geometry content • Students who have completed 7th grade and mastered the content and skills will be prepared for algebra, in 8th grade or after
How to Read the Standards: K-8 • introduction (see page 13)
How to Read the Standards: K-8 • Overview (see page 14)
How to Read the Standards: K-8 • Domains are larger groups of related standards. Standards from different domains may sometimes be closely related. • 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. DOMAIN STANDARD CLUSTER
The high school mathematics standards: Call on students to practice applying mathematical ways of thinking to real world issues and challenges Require students to develop a depth of understanding and ability to apply mathematics to novel situations, as college students and employees regularly are called to do Emphasize mathematical modeling, the use of mathematics and statistics to analyze empirical situations, understand them better, and improve decisions Identify the mathematics that all students should study in order to be college and career ready. Overview of High School Mathematics Standards
How to Read the Standards: High School • Additional mathematics that students should learn in order to take advanced courses such as calculus, advanced statistics, or discrete mathematics is indicated by (+), as in this example: • (+)Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers). • All standards without a (+) symbol should be in the common mathematics curriculum for all college and career ready students. • The high school standards are listed in conceptual categories: • Number and Quantity • Algebra • Functions • Modeling • Geometry • Statistics and Probability
How to Read the Standards: High School • Introduction (p. 62)
How to Read the Standards: High School • Overview (p. 63)
Four Model Course Pathways • A more traditional approach, with two algebra courses and a geometry course and data included in each; • An integrated approach, with three courses that each includes number, algebra, geometry, and data; • A “compacted” version of each pathway that begins in Grade 7 and allows students to study Calculus or other college level courses in high school.
In considering the pathways, there are several things important to note: • The pathways and courses are models, not mandates. They illustrate possible approaches to organizing the content of the CCSS into coherent and rigorous courses that lead to college and career readiness. States and districts are not expected to adopt these courses as is; rather, they are encouraged to use these pathways and courses as a starting point for developing their own. • All college-and career-ready standards (those without a +) are found in each pathway. A few (+) standards are included to increase the coherence of a course. • The course descriptions delineate the mathematics standards to be covered in a course but they are not prescriptions for curriculum or pedagogy. Additional work will be needed to create coherent instructional programs that help students achieve these standards.
Activity: Becoming Familiar with the Standards • Review the Mathematics Standards by marking them up with questions/comments that focus on changes that these standards will require of teachers across all disciplines. • Think through the instructional changes that will arise as a result of the CCSS by talking through the issues that these standards will engender and the problems with resources, including time and the need for professional development.
A TEST THAT IS WORTH TEACHING TO SHOULD…
Math Assessment: Mode of Administration • The through-course components in mathematics will be administered online to students in grade 6 through high school, using an equation editor-type program that allows students to enter responses to mathematical problems via the computer. • Through-course components for grades 3–5 will be administered in paper-and-pencil format because of concerns about young students‘ lack of familiarity with pull-down menus and online mathematics tools. • The Partnership will study the efficacy of online administration of the through-course components to students in grades 3–5 over time. Additionally, as in ELA/literacy, the end-of-year mathematics component will be delivered via computer to students in all grades.
Math-1 and Math-2. Focused Assessments of Essential Topics. • The first two through-course components emphasize standards or clusters of standards (i.e., one to two essential topics) from the CCSS that play a central role during the first stages of mathematics instruction over the school year. • These include standards that are prerequisites for others at the same grade level, as well as standards or clusters of standards for fields of study that first appear during the grade in question. Thus, instead of surveying an overly broad mathematical landscape as typical ―interim assessments‖ currently do, these components will promote the coherent curricular structure embedded in the CCSS. • This approach also will enable the through-course components to provide more useful results to teachers across the range of performance from a blend of one to two brief constructed-response items per topic and one extended constructed-response per topic. • Over time, the Partnership will refine the selection of standards measured by the focused components based on which mathematical topics prove most predictive of success later in the school year.