In reference to the TIMMS study, there is power of the eraser and a gift of time. The Core is asking us to prioritize student and teacher time, to excise out much of what is currently being taught so that we can put an end to the mile wide, inch deep phenomenon that is American Math education and create opportunities for students to dive deeply into the central and critical math concepts. We are asking teachers to focus their time and energy so that the students are able to do the same.
Focus on the math that matters mostFocusing on far fewer topics and treat them with much better care and detail.As shown by the TIMMS study, in the high performing countries there is a relentless focus on specific areas of mathematics ie. addition and subtraction and the quantities they measure at the K-2 level.For the first time, we will model these countries by having fewer topics learned more deeply. These core masteries will lead much fuller level of understanding. In middle and high school, students with this mastery can move on to do work in data and statistics and applying their knowledge to fields such as Algebra, Trigonometry and Calculus. It will also enable them to engage in rich work in modeling multiple representation to other fields such as economics.
We need to ask ourselves – How does the work I’m doing affect work at the next grade level? Coherence is about the scope and sequence of those priority standards across grade bands. How does multiplication get addressed across grades 3-5? How do linear equations get handled between 8 and 9? What must students know when they arrive, what will they know when they leave a certain grade level?
Fluency is the quick mathematical content; what you should quickly know. It should be recalled very quickly. It allows students to get to application much faster and get to deeper understanding. We need to create contests in our schools around these fluencies. This can be a fun project. Deeper understanding is a result of fluency. Students are able to articulate their mathematical reasoning, they are able to access their answers through a couple of different vantage points; it’s not just getting to yes; it’s not just getting the answer but knowing why. Students and teachers need to have a very deep understanding of the priority math concepts in order to manipulate them, articulate them, and come at them from different directions.
The Common Core is built on the assumption that only through deep conceptual understanding can students build their math skills over time and arrive at college and career readiness by the time they leave high school. The assumption here is that students who have deep conceptual understanding can: Find “answers” through a number of different routesArticulate their mathematical reasoningBe fluent in the necessary baseline functions in math, so that they are able to spend their thinking and processing time unpacking mathematical facts and make meaning out of them. Rely on their teachers’ deep conceptual understanding and intimacy with the math concepts
The Common Core demands that all students engage in real world application of math concepts. Through applications, teachers teach and measure students’ ability to determine which math is appropriate and how their reasoning should be used to solve complex problems. In college and career, students will need to solve math problems on a regular basis without being prompted to do so.
This is an end to the false dichotomy of the “math wars.” It is really about dual intensity; the need to be able to practice and do the application. Both things are critical.
Common core dissection 1
Common Core Dissection Staff Development Day Friday, March 23, 2012
DenialThis can’t be happening! I finallyunderstand the 2005 standards!
AngerIt’s unfair! They are changing the curriculum, the test, and the evaluation system at the same time!!
BargainingI promise I’ll be a good teacher, if you will just leave the math curriculum alone.
Depression I don’t care anymore. I’llteach whatever they tell me.
AcceptanceOK-I’m ready to take on the Common Core. Where do I start?
Common Core IntroductionThe State has provided each grade level with its own introduction to the Common Core Learning Standards What do you notice about the focus of the introduction in your grade-level?
Math Practices• On posters around the room, you will find the eight math practices. – These represent the 8 ’habits of mind’ that connect the entire pre-K through 12 continuum.• Please take a few minutes to reflect on each math practice statement and consider – What does it mean? – What does it look like?
1. Make Sense of Problems and Persevere in Solving Them• Read and understand the problem• Organize information to make a plan (more than one approach is acceptable)• Solve and adjust the process as necessary• Evaluate the solution and decide how you will use the solution to answer the question
2. Reason Abstractly and Quantitatively• Take a given situation and represent it symbolically (decontextualization)• Assign meaning to the symbols by putting them back in context (contextualize)
3. Construct Viable Arguments and Critique the Reasoning of Others• Listen to or read the arguments of others• Decide whether the arguments make sense• Ask useful questions to clarify or improve the arguments
4. Model With Mathematics• Apply math concepts to solve problems that arise in every day life• Simplify a complicated situation and make revisions as needed• Use tools such as diagrams, graphs charts to analyze math relationships and draw conclusions• Interpret results and reflect, “Does this make sense?”• Make adjustments, if necessary
5. Use Appropriate Tools Strategically• Consider available tools such as paper/pencil, concrete models, rulers, software• Develop a familiarity with tools• Make sound decisions about which tools to use for the situation• Use technological tools to deepen the understanding of concepts
6. Attend to Precision• Calculations are accurate and efficient• Explanations include clear definitions, correct use of symbols (especially the equal sign), units of measure, axes labels, etc.
7. Look For and Use Structure• Look for patterns and use what is known to apply or extend the pattern• Look for structure by identifying the conceptual way that we think about math• Find and follow the “rules” that mathematicians use to solve problems
8. Look For and Express Regularity in Repeated Reasoning• Notice repetitive actions in counting and computation• Look for shortcuts such as rounding• Continuously check work to see if answers are reasonable• Ask, “Does the answer match the question?”
The Six Shifts in Mathematics• SHIFT 1: FOCUS• SHIFT 2: COHERENCE• SHIFT 3: FLUENCE• SHIFT 4: DEEP UNDERSTANDING• SHIFT 5: APPLICATION• SHIFT 6: DUAL INTENSITY Group Discussion: • What does this shift mean? • What does it look like in the classroom? From the teacher’s perspective? From the students’ perspectives?
Mathematics Shift 1: FocusWhat the Student Does… What the Teacher Does… What the Principal Does…•Spend more time thinking and •Make conscious decisions about •Work with groups of mathworking on fewer concepts. what to excise from the curriculum teachers to determine what•Being able to understand and what to focus content to prioritize most deeplyconcepts as well as processes •Pay more attention to high and what content can be removed(algorithms). leverage content and invest the (or decrease attention). appropriate time for all students to •Determine the areas of intensive learn before moving onto the next focus (fluency), determine where topic. to re-think and link (apply to core •Think about how the concepts understandings), sampling (expose connects to one another students, but not at the same •Build knowledge, fluency and depth). understanding of why and how we •Determine not only the what, but do certain math concepts. at what intensity. •Give teachers enough time, with a focused body of material, to build their own depth of knowledge.19
An Examination of Focus• The standards were Intense focus Rethink and link Sampling written to reflect a 70% 20% 10% deepening of key K-2 Addition and Geometry and Patterns Subtraction Measurement Statistics and Data understandings at Concepts Probability certain grade levels. Skills Estimating Arithmetic Problem Solving 3-5 Multiplication and Area Patterns• There are 2-4 focal areas Division of Whole Volume Statistics and Data Numbers and Fractions Probability per grade-level. A good balance of o Concepts o Skills• To ensure the degree of o Problem Solving understanding and 6-8 Proportional Reasoning Quantitative Statistics application & Linearity Relationships & required, more time and Algebra Functions Geometric practice will need to be Measurement devoted to these particular concepts.
Priorities in Math Priorities in Support of Rich Instruction and Expectations ofGrade Fluency and Conceptual Understanding Addition and subtraction, measurement usingK–2 whole number quantities Multiplication and division of whole numbers and3–5 fractions Ratios and proportional reasoning; early 6 expressions and equations Ratios and proportional reasoning; arithmetic of 7 rational numbers 8 Linear algebra 21
Mathematics Shift 2: CoherenceWhat the Student Does… What the Teacher Does… What the Principal Does…•Build on knowledge from •Connect the threads of •Ensure that teachers ofyear to year, in a coherent math focus areas across the same content acrosslearning progression grade levels grade levels allow for •Think deeply about what discussion and planning to you’re focusing on and the ensure for ways in which those focus coherence/threads of main areas connect to the way it ideas was taught the year before and the years after22
Mathematics Shift 3: FluencyWhat the Student Does… What the Teacher Does… What the Principal Does…•Spend time practicing, •Push students to know •Take on fluencies as awith intensity, skills (in high basic skills at a greater stand alone CC alignedvolume) level of fluency activity and build school •Focus on the listed culture around them. fluencies by grade level •Create high quality worksheets, problem sets, in high volume23
The Special Case of Fluency• Fluency is more than just “knowing facts by memory”• It is the fast and accurate completion of a continuum of mathematical operations and procedures• Each grade level has 1-2 key fluency goals
Mathematics Shift 4: Deep UnderstandingWhat the Student Does… What the Teacher Does… What the Principal Does…•Show, through numerous •Ask yourself what •Allow teachers to spendways, mastery of material mastery/proficiency really time developing their ownat a deep level looks like and means content knowledge•Use mathematical •Plan for progressions of •Provide meaningfulpractices to demonstrate levels of understanding professional developmentunderstanding of different •Spend the time to gain the on what student masterymaterial and concepts depth of the understanding and proficiency really •Become flexible and should look like at every comfortable in own depth grade level by analyzing of content knowledge exemplar student work25
Mathematics Shift 5: ApplicationWhat the Student Does… What the Teacher Does… What the Principal Does…•Apply math in other •Apply math including •Support science teacherscontent areas and areas where its not directly about their role of mathsituations, as relevant required (i.e. in science) and literacy in the science•Choose the right math •Provide students with real classroomconcept to solve a problem world experiences and •Create a culture of mathwhen not necessarily opportunities to apply application across theprompted to do so what they have learned school26
Mathematics Shift 6: Dual IntensityWhat the Student Does… What the Teacher Does… What the Principal Does…•Practice math skills with •Find the dual intensity •Provide enough mathan intensity that results in between understanding class time for teachers tofluency and practice within focus and spend time on•Practice math concepts different periods or both fluency andwith an intensity that different units application offorces application in novel •Be ambitious in demands concepts/ideassituations for fluency and practice, as well as the range of application27
Highlight• In one color, highlight the content that we currently teach• In another color, highlight the content that is NEW to our grade level
An example from 4th gradeSkill (verb) Content (noun) Method (using…)Recognize Relationship between digits (any place is 10 times the digit to its right)Read Multi-digit whole numbers Base-ten numerals, number names, expanded formCompare Multi-digit whole numbers Place value
Let’s Continue• We will break into groups• Each group will complete a portion of the Skills-Content-Method activity• Be ready to share out with the entire grade-level group
Resourceshttp://schools.nyc.gov/Academics/CommonCoreLibrary/SeeStudentWork/default.htmCCLS aligned Performance tasks and instructional supports (units)http://illustrativemathematics.org/standards/k8A small but growing list of sample activities for the CCLShttp://commoncoretools.wordpress.com/category/progressions/Some still in draft form but its getting there!http://ime.math.arizona.edu/commoncore/Bring together some of the best math minds…http://insidemathematics.org/index.php/classroom-video-visitsA collection of videos, open-ended problem solving taskhttp://nymathstandards.pbworks.com/w/page/45609650/Unpacking%20the%20Content%20Standards%20for%20MathNYS math teacher-created break down of each standard