1. A Guide to the K-8 and High School Publishers’ Criteria for the CCSSM
Dana Cartier
ISBE Content Area Specialist
Slides adapted from the K-8 Publishers’ Criteria
Map
Take a page of textbook and find information and cut stuff out.
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2. 3.OA.9 Task #1 Fill in the blank below and explain. 2, 4, 6, ____
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3. 3.OA.9 Task #2
In each column and each row of the table even and odd numbers alternate. Explain why.
Explain why the diagonal, from top left to bottom right, contains the even numbers 2,4,6,8, and 10.
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4. What were the similarities and differences of the previous tasks?
Compare the Tasks
What were the similarities and differences of the previous tasks?
Compare
Scale
Blank Venn Diagram
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5. Objectives Explore the math criteria
Introduce the K-8 & High School Publishers’ Criteria
Learn the why and how of using the criteria
Explore the math criteria
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6. What is the Publishers’ Criteria?
Supports faithful implementation of
the CCSSM
Developed by the three lead authors
Phil Daro, Bill McCallum, Jason
Zimba
Two different documents- similiar
The Publishers’ Criteria was published July 20, 2012 and was created by the three lead authors of the CCSSM. It was designed to encourage faithful implementation that supported the spirit of the CCSSM.
While this document says it is a K-8 document, it makes mention of high school and also suggests that the criteria will also be applicable in the upper grade levels. It is expected that an updated version of this will come out in 2013 with high school criteria.
A tool designed to support faithful implementation of the CCSSM
Developed by the three lead authors of the CCSSM
Intended for K-8, but many comments are high school applicable
Updated and released in early April, Included a High School document for the first time.
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7. Why Do We Need This?
Standards cannot raise achievement. Material should connect to assessment. Educators complain about what is missing, but not about what has crept in.
Chart going up
“Standards by themselves cannot raise achievement. Standards don’t stay up late at night working on lesson plans, or stay after school making sure every student learns” as it says on Page 1.
When writing the CCSSM, the authors found that people were pleased with what they saw but wanted to add “just one more thing”, this is how textbooks across the country kept getting bigger and bigger.
“Standards by themselves cannot raise achievement” page 1
Educators complain about what is missing, but not about what has crept in.
“Because conventional textbook coverage is so fractured, unfocused, superficial, and unprioritized, there is no guarantee that most students will come out knowing the essential concepts of algebra.” Wiggins, 2012
Connecting material to assessment
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8. Wiggins, 2012
Because conventional textbook coverage is so fractured, unfocused, superficial, and unprioritized, there is no guarantee that most students will come out knowing the essential concepts of algebra.
Fracture
Textbook
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9. How Can We Use This Tool
Inform purchase and adoption Work with previously purchased materials Review and guide teacher-developed materials Plan professional development
Ladder
Page 6
Informing purchases and adoptions
Test claims of alignment
Working with previously purchased materials
Plan to modify or combine existing resources
Reviewing and guiding teacher-developed materials
Professional development
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10. Focus Coherence Rigor Part I: Major Shifts Rock shifts
The key shifts in the CCSSM
Focus - Even when state standards become more focused, instructional materials do not because they have to cover all states.
Coherence – material was not organized so that math made sense.
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11. Focus Math “needs to lose a few pounds” (p.3)
Consider what is not said
Math “needs to lose a
few pounds” (p.3)
Differentiation
“Cannot add “just one more thing”
Extensions
Phil Daro: it is better to inadvertently leave something out than to teach too much
75% of time is spent on the major work, especially the first half of the year
At least 70% of the PARCC assessment will come from the Major clusters.
Part 1 focuses on the key shifts in the CCSSM. The first key shift is focus. One of the major problems with earlier standards is that they lacked focus. The CCSSM says not only must the standards provide focus, but so should the curricular materials. K-5 should focus on arithmetic with other topics complementing the concepts, skill and applications of it. Where material has been moved to a higher grade, the textbook should not include it in the lower grades.
Good stuff may have been lost, but the sacrifice is higher learning of the material still existing.
When teachers teach fewer topics, students do better on tests even if they have not “covered” everything on the test.
Grade-by-grade progressions mean that students are working on grade-level work early in the school work.
Differentiation is sometimes necessary, but materials often manage unfinished learning from earlier grades inside grade-level work, rather than setting aside grade-level work to reteach earlier content.
In K-8 the material should devote at least 65% and up to approximately 85% of the class time to the major work of the grade with Grades K-2 nearer the upper end of the range.
High School should focus on widely applicable prerequisites. Material should coherently include all of the standards in High School (without the + standards), with a majority of the time devoted to building the particular knowledge and skills that are most applicable and prerequisite to a wide range of college majors and postsecondary programs. Materials developed to prepare students for STEM majors ensure that STEM-intending students learn all of the prerequisites in the Standards necessary for calculus and other advanced courses.
Grade-level work begins during the first two to four weeks of instruction
“Teach less, learn more”
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12. Math should make sense Coherence A progression of learning
Practice-content coherence
A progression of learning
Coherence supports focus
Use supporting material to teach major work
The second shift is coherence, that the material that is being presented in materials should relate and connect. “Math should make sense” Supporting topics such as bar graphs and statistics are used as an application to integrate arithmetic operations. Therefore the coherence is supporting the focus. The standards are not separate events but an interrelated mathematical system.
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13. Rigor Balance with equal intensity Conceptual understanding
Procedural skill and fluency
Application
In the past, rigor has been somewhat specialized. The math wars had two sides. One argued that math was only about the skill and the answer with the other side argued that it was all about the concept and the process. The common core came along and said that it is about the balance, not an either/or. Unfortunately many textbooks also focused on rigor in one area but not necessarily the others.
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14. Conceptual Understanding
“Understand” High-quality conceptual problems Elicit conversation Identify relationships Multiple representations
The CCSSM uses some form of the work understand 263 times. Students need to use the practice standards to demonstrate understanding by being explain to explain their work and recognize multiple representations. Curricular material needs to include high quality conceptual tasks.
Curricula have not always been balanced
Some stress fluency in computation but not conceptual understanding
Some stress conceptual understanding without fluency
Some focus on pure math without applications
Some focus on application without acknowledging math does not teach itself
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15. Fluency
“Fluently” Methods are based on mathematical principles, not mnemonics or tricks Quickly and accurate, to work with flow. It isn’t halting, stumbling, or reversing oneself.
Fluency starts from understanding of the process and then after giving the students opportunities to experience the concept in multiple ways. Whenever the word fluently appears in a content standard, the word means quickly and accurate, to work with flow. Fluent isn’t halting, stumbling or reversing oneself.
Manipulatives used to enhance understanding but phased out before reaching fluency
Materials use fluency only where required in standards
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16. Fluency
ADD LINK
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17. Fluency Example
2. Fill in the blanks below to make the equation true for every value of x. Explain the steps you took, as well as any math properties you used.
1. Write a simplified expression for the following:
7r + 2(r + 3)
Think, pair, share. 3 minutes think, 3 minutes pair, 3 minutes share.
“Fluently”
Methods are based on mathematical principles, not mnemonics or tricks
Fluency starts from understanding of the process and then after giving the students opportunities to experience the concept in multiple ways. Fluent isn’t halting, stumbling or reversing oneself. It is quickly and accurate, to work with flow.
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18. Application Single-step and multi-step contextual problems
Require students to make assumptions in order to model a situation
Modeling – a practice standard and a HS Conceptual Category
Application must also be balanced, focusing on where real-world is mentioned in the standards. Mathematical Practice 4: Model with Mathematics also suggests using application frequently.
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19. What is the mathematical question?
Application Problem
My phone beeps with a text from
my friend who is an event coordinator:
My producer sent me only 50 feet of red velvet rope and 4 poles! I don’t know what he was thinking, how can I fit all of the VIPs in this section?“
What is the mathematical question?
Example of contextual problems.
Modeling – a practice standard and a HS Conceptual Category
Precisely answer the question that we wrote.
Write a text to the event coordinator to explain the answer and to explain how he could always do similar problems in the future.
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20. Application How many dinosaurs are standing in the lake?
Example of Students make assumptions in order to model a situation.
How many dinosaurs are standing in the lake?
Explain how you know. Use words and mathematical language to explain your solution.
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21. Part II: Criteria for Materials and Tools
10 criteria to analyze mathematics curricular materials 1 criterion to align mathematics to other disciplines
Major work should be huge focus in first half of the year.
Progression to middle school algebra is provided on page 8.
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22. K-8 Publishers’ Criteria
Part II: Criteria for Materials and Tools Aligned to the K-8 Standards
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23. 1. Focus on Major Work 65%-85%of time is spent on the major work
Better to inadvertently leave something out than to teach too much
65%-85%of time is spent on the major work
Magnifying glass
The critical areas are emphasized in the introduction to each grade in K-8 of the CCSSM. This should be the focus of the school year with most of the other standards supporting this work.
Phil Daro says it is better to inadvertently leave something out than to teach too much
65-85% of time is spent on the major work, especially the first half of the year
Critical Areas in K-8 and the Major clusters in PARCC Model Content Framework
At least 70% of the PARCC assessment will come from the Major clusters.
In K-8 the material should devote at least 65% and up to approximately 85% of the class time to the major work of the grade with Grades K-2 nearer the upper end of the range.
High School should focus on widely applicable prerequisites. Material should coherently include all of the standards in High School (without the + standards), with a majority of the time devoted to building the particular knowledge and skills that are most applicable and prerequisite to a wide range of college majors and postsecondary programs. Materials developed to prepare students for STEM majors ensure that STEM-intending students learn all of the prerequisites in the Standards necessary for calculus and other advanced courses.
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24. 5th Grade PARCC Model Content Framework
Here is an example of the 5th grade PARCC Model Content Framework. Clusters are labeled as major, supporting and additional and the assessments will be in relation to this.
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25. 2. Focus in Early Grades
Don’t cover material early Don’t assess material early Patterns only within arithmetic
Young kids
In the past, because states had standards at different grade levels, curricular material included these topics in all of these grade levels. Because of this, and broad state standards, material was covered and assessed before the students were ready for it. Teachers need to let go, even if it means losing that “fun” or “perfect” lesson and realize that the material that has moved up needs to be let go.
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26. In the past probability was in 1st-12th grade and didn’t change much.
Tell the 1st grade lesson study story on probability.
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27. 3. Focus and Coherence through Supporting Work
Supporting work enhances major work Supporting work does not detract from focus
Scaffold
The supporting work gives context for the major work and is not separate.
For example, materials for K-5 generally treat data displays as an occasion for solving grade-level word problems using the four operations (3.MD.3); materials for 7th grade take advantage of opportunities to use probability to support ratios, proportions, and percents.
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28. 2nd Grade Example from Measurement and Data Progression
The measurement and data progression document gives a 2nd grade example. The question here asks the students to do subtraction, not just stating what is there. The students also begin the work of pre-multiplication leading towards third grade.
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29. 4. Rigor and Balance Conceptual Understanding Fluency Application
Three aspects are not always together.
Three aspects are not always separate.
All 3 for comprehensive material and at least one for supplemental or targeted resources.
Rigor needs to be balanced in the areas of conceptual understanding, fluency and application.
Conceptual understanding: material amply feature high-quality conceptual problems and questions. This includes brief conceptual problems with low computational difficulty (“Find a number greater than 1/5 and less than ¼”); brief conceptual questions (“If the divisor does not change and the dividend increases, what happens to the quotient?”); and problems that involve identifying correspondences across different mathematical representations of quantitative relationships.
Procedural Skill and Fluency: Purely procedural problems and exercises are present. These include cases in which opportunistic strategies are valuable (the sum or the system x+y=1, 2x+2y=3) as well as ample number of generic cases so students can learn and practice efficient algorithms.
Application: ample number of single-step and multi-step contextual problems include problems in which students must make their own assumptions or simplifications in order to model a situation mathematically. Individual and group work. Modeling builds slowly across k-8, and applications are relatively simple in earlier grades.
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30. 5. Consistent Progressions
Grade-by-grade Grade-level problems Relating to prior knowledge
grade-by-grade: students make tangible progress during each given grade, as opposed to substantially reviewing then marginally extending from previous grade.
Grade-level: materials often manage unfinished learning from earlier grades inside grade level work. Example: the development of fluency with division using the standard algorithm in grade 6 is the occasion to surface and deal with unfinished learning about place value. Students ready for more can be provided with problems that take grade-level work in deeper directions.
Relating: prior knowledge becomes reorganized and extended to accommodate new knowledge. Material makes extensions of prior knowledge explicit (basic ideas of place value then extend across the decimal point to tenths and beyond, properties of operations with whole numbers, then extend them to fractions, variables, and expressions.
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31. 6. Coherent Connections
Curricular material makes connections between clusters and domains CCSSM are more than a sum of their parts
Include learning objectives that are visibly shaped by CCSSM cluster headings. Cluster headings state the point of, and lend additional meaning to, the individual content standards that follow. Also can signal multi-grade progression (“apply and extend previous understanding…”)
Including problems and activities that serve to connect 2 or more clusters in a domain, or 2 or more domains in a grade, in cases where the connections are natural and important. Ex: robust work in 4.NBT should sometimes or often synthesize across the clusters listed in that domain; robust work in grade 4 should sometimes or often involve students applying their developing computation NBT skills in the context of solving word problems detailed in OA.
Preserving the focus, coherence, and rigor of the standards even when targeting specific objectives. Some standards are compound statements. It is sometimes helpful or necessary to isolate part of a compound standard, but not always, and not at the expense of the Standards as a whole.
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32. 7. Practice-Content Connections
Not separate Present throughout Grounded in the content standards Accompanying teacher-support
In my opinion, the connection of practice to content standards should be one of the first criteria. The practice standards are not a separate document they are integrated throughout the school year and throughout the content standards. They are not separate but are grounded in the work of the content standards. Curricular materials should give examples and discussions of how these connections can happen.
Teacher support to lead discussion focused on the practice standards
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33. 8. Focus and Coherence via Practice Standards
Connect practice and content standards as specified
Structure for structural themes (MP7)
Using repetition to find mathematical regularity (MP8)
Content and practice standards are not connected mechanistically or randomly, but instead support focus and coherence.
MP.7- connect looking for and making use of structure with structural themes emphasized in the standards such as properties of operations, place value decompositions of numbers, numerators and denominators, numerical and algebraic expressions, etc.
MP.8- use repeated reasoning to explore content that is emphasized in the standards. K-5 regularity in repeated reasoning to shed light on the 10x10 addition table, and 10x10 multiplication table, properties of operations, relationship between addition and subtraction or multiplication and division, and the place value system. 6-8 to shed light on proportional relationships and linear functions.
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34. 9. Careful Attention to Each Practice Standard
The practice standards have to be implemented faithfully as well. The entire standard needs to be considered “Make sense of problems and persevere in solving them” includes multiple components. Students must decipher information from a problem and they must persevere beyond the point where they may want to give up. This may include multi-step problems.
MP1 Persevere beyond the point that they would like to give up
MP5 Students deciding what tools to use
Full meaning and spirit of the entire practice standard
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35. 10. Emphasis on Mathematical Reasoning
Construct viable arguments Problem-solving as argument Specialized language
Argument Picture (lawyer)
Explained more in next three slides.
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36. Construct Viable Argument
25-50% of students’ time
Independent thinking, classroom discussion and written work
Critique arguments, error analysis
Multi-step problems
Student-devised strategy
Cohesive arguments that can be verified and critiqued
Not a jumble of steps
Same picture
Students need to spend a large portion of their time communicating their reasoning. Materials and class time should give opportunity for independent, group and written explanations. Materials should not place this in a position where it can be considered optional. i.e. Students can spend time doing error analysis on sample problems from the materials or from classmates.
25-50% of students’ time spent communicating mathematical reasoning
Independent thinking, classroom discussion and written work
Not optional or avoidable
Focused on content and major work
Critique given arguments, error analysis of real or fictitious student work
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37. Specialized Language
“The language of argument, problem-solving and mathematical explanations are taught rather than assumed.”
Language of representations
Diagrams, tables, graphs, symbolic expressions, drawing, images, text
Helpful for ELL students
Dictionary
Students may need to learn how to talk in the mathematical language. Using multiple representations can help bridge language gaps. ELL students may be able to visually explain a process and overtime learn to communicate this verbally.
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38. High School Publishers’ Criteria
Part II: Criteria for Materials and Tools Aligned to the High School Standards
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39. 1. Focus on Widely Applicable Prerequisites
Material should coherently include all of the standards in High School (without the + standards), with a majority of the time devoted to building the particular knowledge and skills that are most applicable and prerequisite to a wide range of college majors and postsecondary programs
High School should focus on widely applicable prerequisites. Material should coherently include all of the standards in High School (without the + standards), with a majority of the time devoted to building the particular knowledge and skills that are most applicable and prerequisite to a wide range of college majors and postsecondary programs. Materials developed to prepare students for STEM majors ensure that STEM-intending students learn all of the prerequisites in the Standards necessary for calculus and other advanced courses.
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40. Page 8- High School Publishers’ Criteria
Table 1 lists clusters and standards with relatively wide applicability across a range of postsecondary work. Table 1 is a SUBSET of the material students must study to be college and career ready (CCSSM, pp. 57, 84)
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41. 2. Rigor and Balance Conceptual Understanding Fluency Application
Three aspects are not always together.
Three aspects are not always separate.
All 3 for comprehensive material and at least one for supplemental or targeted resources.
Rigor needs to be balanced in the areas of conceptual understanding, fluency and application.
Conceptual understanding: material amply feature high-quality conceptual problems and questions. This includes brief conceptual problems with low computational difficulty (“What is the maximum value of the functions f(t)=5-t^2”); brief conceptual questions (“Is the sqrt(2) a polynomial? How about ½(x+sqrt2)+1/2(-x+sqrt2)?’); and problems that involve identifying correspondences across different mathematical representations of quantitative relationships. (equation and graph, solving equations as a process of answering a question, analyzing a non-linear equation f(x)-g(x) by graphing f and g disticntly on a single axis, etc.
Procedural Skill and Fluency: algebra is the language of much of mathematics. Sufficient practice with algebraic operations is provided so as to make realistic the attainment of the Standards as a whole (fluency in algebra can help students get past the need to manage computational details so they can observe the structure MP.8). Interwoven with conceptual understanding. Purely procedural problems and exercises are present. These include cases in which opportunistic strategies are valuable ((3x-2)^2=6x-4) as well as ample number of generic cases so students can learn and practice efficient and general methods (solving c+8-c^2=3(c-1)^2-5).
Application: ample number of single-step and multi-step contextual problems. problems in which students must make their own assumptions or simplifications in order to model a situation mathematically. Individual and group work. Modeling is a Math Practice standard and also a content category. Materials include ample number of high school level problems that involve applying key takeaways from K-8.
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42. 3. Consistent Content Base course on content
Give students extensive work with course-level problems
Explicitly relates course content to prior knowledge
Basing courses on the content specified in the Standards: Materials do not create new standards or rewrite standards. Does not introduce gaps in learning by omitting any content.
Give all students extensive work with course-level problems: Previous grade or course review is clearly identified as such. Unfinished learning from earlier grades and course s is normal and prevalent. The equation of a circle is an occasion to surface and deal with unfinished learning about correspondence between equations and their graphs.
Relating course level concepts explicitly to prior knowledge from earlier grades and courses: They learn basic ideas of functions and then extend them to deal explicitly with domains. They learn about expressions as recording calculations with numbers then extend them to symbolic objects in their own right.
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43. 4. Coherent Connections
Curricular material makes connections between clusters and domains CCSSM are more than a sum of their parts
Include learning objectives that are visibly shaped by CCSSM cluster headings. Cluster headings state the point of, and lend additional meaning to, the individual content standards that follow. Also can signal multi-grade progression (“apply and extend previous understanding…”)
Including problems and activities that serve to connect 2 or more clusters in a domain, or 2 or more domains in a grade, in cases where the connections are natural and important. Ex: creating equations (A.CED) isn’t very valuable in itself unless students can also solve them (A.REI). A.REI.11 connects functions to equations in a graphical context.
Preserving the focus, coherence, and rigor of the standards even when targeting specific objectives. Some standards are compound statements. It is sometimes helpful or necessary to isolate part of a compound standard, but not always, and not at the expense of the Standards as a whole.
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44. 5. Practice-Content Connections
Not separate Present throughout Grounded in the content standards Accompanying teacher-support
In my opinion, the connection of practice to content standards should be one of the first criteria. The practice standards are not a separate document they are integrated throughout the school year and throughout the content standards. They are not separate but are grounded in the work of the content standards. Curricular materials should give examples and discussions of how these connections can happen.
Teacher support to lead discussion focused on the practice standards
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45. 6. Focus and Coherence via Practice Standards
Connect practice and content standards as specified
Structure for structural themes (MP7)
Using repetition to find mathematical regularity (MP8)
Content and practice standards are not connected mechanistically or randomly, but instead support focus and coherence.
MP.7- connect looking for and making use of structure with structural themes emphasized in the standards such as purposefully transforming expressions, linking the structure of an expression to a feature of its context, grasping the behavior of a function defined by an expression, etc.
MP.8- use repeated reasoning to explore content that is emphasized in the standards shed light on algebra and functions by summarizing repeated numerical examples in the form of equations or in the form of recursive expressions that define repeated reasoning with the slope formula to writing equations for straight lines in various forms, rather than relying on memorizing all those forms in isolation.
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46. 7. Careful Attention to Each Practice Standard
The practice standards have to be implemented faithfully as well. The entire standard needs to be considered “Make sense of problems and persevere in solving them” includes multiple components. Students must decipher information from a problem and they must persevere beyond the point where they may want to give up. This may include multi-step problems.
MP1 Persevere beyond the point that they would like to give up
MP5 Students deciding what tools to use
Look for and express meaning (MP. 8) repeated calculations must lead to an insight (when I substitute x-k for x in a function f(x), where k is any constant, the graph of the function shifts k units to the right.
Full meaning and spirit of the entire practice standard
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47. 8. Emphasis on Mathematical Reasoning
Construct viable arguments Problem-solving as argument Specialized language
Argument Picture (lawyer)
Explained more in next three slides.
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48. Connecting Criterion: Consistency with CCSSM
Materials for science and technical subjects are consistent with CCSSM Grade-level appropriate mathematics in other disciplines
Science Physics
The last criterion is not for mathematics curricular material. It states that materials in other disciplines should be consistent with math. Science and technical subjects should not be using mathematics that has not yet been covered in the mathematics curriculum. This was cause haphazard rushed learning to be attempted, possibly causing confusion when the students actually learn the material in the context of mathematics. These subjects should, however, encourage the application of the math that has been learned to their subjects, when possible grade-level appropriate materials. Publishers should consider this when creating curricular materials for all subjects.
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49. SHORT Indicators of Quality Separate teacher material
Problems are worth doing
Separate teacher material
Support for diverse learners
Indicators of Quality
SHORT
Olympic gold
Curricular materials include problems that are worth doing. Each problem has a purpose, whether that be taking the math to the next step or applying what they already know. Problems involve something the students do not know, while exercises have students follow the same steps they have already learned on additional problems. Both in class and at home assignments are designed to progress student thinking with intention.
Problems are worth doing
Problems vs. exercises
Each problem has a purpose
Assignments are purposefully designed
Intentional sequence
Variety of student products
Variety in the pacing of each standard
Variety in the pacing of each standard
Best practices with manipulatives
Variety of student products
Visual design is clear
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50. Avoid crosswalks Don’t take the common out of COMMON core
Align to the letter and spirit of the standards
Don’t take the common
out of COMMON core
Map
The criteria states that the point is not to do crosswalks, but to look for the spirit of the standards. Using this and the CCSSM, decide what needs to be taken out of the curriculum to allow more time for the focus topics. Work with the common core and the standards and don’t spend time rewording the standards.
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51. Dana Cartier ISBE Content Area Specialist Area 5&6 dcartier@stclair
Dana Cartier ISBE Content Area Specialist Area 5&6
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