Fostering Friendships - Enhancing Social Bonds in the Classroom
Blue ribbon
1. Common Core Mathematical
Practices, Content and Tasks
April 12, 2013
Presented by:
Kathy Favazza, Reading Public Schools
Brenda Regan, Tewksbury Public Schools
Massachusetts PARCC Leader Fellow 2012-2014
2. Session Focus
Strengthen expertise in and develop
a common understanding of
•Common Core State Standards for
Mathematic Practices
•Connect Practice Standards, Content
Standards and Assessment
3. Common Core State Standards
Timeline for P-16 Alignment of MA Standards
and Assessments in ELA/Literacy and Math
2012-2013 2013-2014 2014-2015
Near full Full Full implementation
implementation of implementation of of new assessments
ELA and math ELA and math of ELA and math
standards standards standards
Implement balance of Monitor curriculum Administer
changes in curriculum and instruction fully assessments fully
and instruction aligned to the 2011 aligned to the 2011
standards standards
4. Common Core State Standards
Math MCAS Transition Schedule, Grades 3-8
http://www.doe.mass.edu/mcas/transiti
on/?section=math3-8
6. The New PARCC Assessment
PARCC Timeline
SY 2012-13 SY 2013-14 SY 2014-15 Summer 2015
SY 2010-11 SY 2011-12
First year Second year Full Set
pilot/field pilot/field administratio achievement
Launch and Development
testing and testing and n of PARCC levels,
design phase begins
related related assessments including
research and research and college-ready
data collection data collection performance
levels
7. Diagnostic Assessment
•Early indicator of student
knowledge and skills to inform
The New PARCC Assessment
instruction, supports, and PD
•Non-summative
Assessment Design: English Language
Arts/Literacy and Mathematics, Grades 3-11
2 Optional Assessments/Flexible Administration
Mid-Year Assessment Performance-Based End-of-Year
•Performance-based Assessment (EOY)
Assessment (PBA)
•Emphasis on hard-to- •Extended tasks •Innovative, computer-
measure standards •Applications of based items
•Potentially summative
concepts and skills •Required
•Required •Summative
•Summative
Speaking And Listening Assessment
•Locally scored…ELA only at this time.
•Non-summative,…yet required
9. Claims Driving Design: Mathematics
Master Claim: On-Track for college and career readiness.
Students solve grade-level/course-level problems in mathematics as set forth in the
Standards for Mathematical Content with connections to the Standards for
Mathematical Practice.
Sub-claim B
Sub-claim A: Sub-claim C
Students solve problems
Students solve problems Students express
involving the additional and
involving the major content mathematical reasoning by
supporting content for their
for their grade level with constructing mathematical
grade level with connections to
connections to practices arguments and critiques
practices
Sub-claim E
Sub-claim D
Students demonstrate fluency
Students solve real world
in areas set forth in the
problems engaging particularly
Standards for Content in grades
in the modeling practice
3-6
11. Mathematical Practices
Brainstorm a list of:
What does the practice standard mean?
How do students apply practices to content?
How do the practices look in the classroom?
12. Mathematical Practices Gallery Walk
With your group you will visit the other
Mathematical Practices:
Read the responses.
Add any additional descriptors.
What does the practice standard mean?
How do students apply practices to content?
How do the practices look in the classroom?
13. Practices Content
• Teaching content without practices is
possible.
• Teaching practices without content is
NOT possible.
• Practices should be connected to
specific content.
14. Practice-Integrated Standard = Explicit
• 3.NF.3 Recognize and generate simple equivalent
fractions. Explain why the fractions are equivalent.
• F-BF.2 Write arithmetic and geometric sequences
both recursively and with an explicit formula, use
them to model situations, and translate between the
two forms.
15. Practice-Related Standard = Implicit
• 4.NBT.5 Multiply a whole number of up to four digits by
a one-digit whole number, and multiply two two-digit
numbers, using strategies based on place value and the
properties of operations. Illustrate and explain the
calculation by using equations, rectangular arrays,
and/or area models.
16. Three Types of Standards
• Practice-Integrated Standard = Explicit
• Practice-Related Standard = Implicit
• Content standards not associated with a
particular practice
18. Practice-Forward Tasks
•A practice-forward task is one that aligns to a
practice-integrated or practice-related content
standards.
•A practice-forward task is one in which it is
unlikely or impossible to earn full credit on the
task without engaging in the practice.
19. Practice-Forward Tasks
• Algebraic word problems in which success
depends on carefully defining variables
• Attend to Precision (Mathematical Practice 6)
20. Standards for Mathematical
Practice Assessment
Where? What kind?
•Performance Based Assessment •Practice-Integrated Tasks
(PBA) •Practice-Related Tasks
•End-of-Year Assessment (EOY)
Which ones? How many?
•Different ones with different •At least one per content domain
frequencies
22. Mathematics Master Claim:
On-Track for college and career readiness.
Sub-claim A: Students
solve problems
involving the major
content for their grade
level with connections
to practices .
24. 3rd Shift: Rigor Dr. Core, the book seems to
Dr. Parcc, do you include procedural skills that
see any of the 3 lead to fluency, but I really
indicators that don’t see much application and
rigor has set in? even less building of
conceptual understanding.
25. 快
rá 速 một
cách
、
pi nha
• Quickly and Accurately
准 nh
Fluently
da
m 确 chó
• How can we become fluent?
e 地
ng
và
nt chín
h
e xác
y
ac
cu
ra
at
el
y
26. Fluency and Culminating
Standards Sub-claim E
• Fluency Expectations
(K-8) Students
• Fluency demonstrate
Recommendations (9- fluency in areas set
11) forth in the Standards
• Culminating Standards for Content in grades
3-6.
27. Mathematics Master Claim:
On-Track for college and career readiness.
Students solve grade-level/course-level problems in mathematics
as set forth in the Standards for Mathematical Content
with connections to the Standards for Mathematical Practice.
Sub-claim A: Sub-claim B Sub-claim C
Students solve problems Students solve problems Students express
involving the major content involving theadditional and mathematical reasoning by
for their grade level with supporting content for constructing mathematical
connections to practices. their grade level connections
with arguments and critiques
to practices.
Sub-claim D Sub-claim E
Students solve real world Students demonstratefluency
problems engaging in areas set forth in the
particularly in the modeling Standards for Content in
practice. grades 3-6.
27
Hinweis der Redaktion
The focus of our math-specific break-out sessions will be to: Strengthen our individual expertise in, and develop a common understanding of, <*>the Common Core State Standards for Mathematics and <*>the PARCC Model Content Frameworks You will be participating in some focused activities that will Model how you, the cadre members, can replicate these activities with audiences in your respective states. (Many of you are undoubtedly already very knowledgeable about these topics, but some of your potential audiences may not be. Remember that we are modeling activities that you can use with less “elite” audiences than this one :>) That is a nice, succinct description of our agenda for the next few days. These two documents, The Common Core State Standards and the PARCC Model Content Frameworks, are the foundation for the development of the PARCC assessment.
… the alignment and assessment timeline…?
How do you know what standards are imperative as we go through the next 2 years? Look at the MCAS Transition Schedule…
… the alignment and assessment timeline…?
The testing schedule is 1 piece of the CCSS roll out. Please note K-2 diagnostic assessments are in the developmental phase.
The PARCC assessments have an evidence-centered design. In this design, claims are stated before the test is created. Tasks that students complete on the test provide evidence for the claims. Therefore the claims are the foundation for the assessment system. Let’s take a look at the claim for the Mathematics.
Here you will find the PARCC Master Claim for Mathematics. The PARCC assessment is based on the master claim: “ Students are on track or ready for college and careers.” In understanding the claim we need to consider both: The Common Core State Standards for Mathematical Practice The Common Core State Standards for Mathematical Content We will begin by investigating how the Mathematical Practices are embedded in the sub-claims. The second sentence of the Master claim states: Students solve grade-level/course-level problems in mathematics as set forth in the Standards for Mathematical Content with connections to the Standards for Mathematical Practice. For a short time, let’s focus on those bolded words that draw our attention to the Standards for Mathematical Practice. Posted around the room you will see the 8 Standards for Mathematical Practice. Take just a moment to read these 8 Practice Standards. Extension (or explanation or elaboration) of the Master Claim is shown in the 5 statements called sub-claims. <*> The Practices are explicitly referred to in the first 2 sub-claims. Students will solve problems “with connections to practices.” The topic of major content and additional and supporting content will be discussed in the next session. <*>The 3rd sub-claim refers specifically to mathematical reasoning, which is the focus of Mathematical Practice 3 (Construct viable arguments and critique the reasoning of others). As students construct arguments, they must attend to precision with their use of vocabulary in the statements. The attention to precision is Mathematical Practice 6. <*>The 4th sub-claim specifically mentions “modeling practice,” the focus of Mathematical Practice 4 (Model with mathematics). When students engage in modeling to solve appropriately difficult problems, they are likely to engage in one or more of the remaining Mathematical Practices. <*>The last sub-claim is for grades 3-6 only and focuses on fluency, which can only be developed by engaging in many mathematical practices. We will examine fluency in a later session.
Let’s look deeper into the Standards for Mathematical Practice as we seek to develop a common understanding. Just what are mathematical practices? They are <*>habits of mind. What does the word Habit imply? (automatic, ingrained, developed over time and practice) In math, it means students work with expertise when solving math problems -- which means they do not require prompting. The Practice Standards interact with and overlap each other. They are not a checklist and can not be evaluated in isolation. Several different practices are often times used together in solving a problem. The Practice Standards describe the kind of mathematics teaching and learning that should be fostered in the classroom. They must be embedded in instruction, discussions, and activities. To promote such an environment, students should have opportunities to work on carefully designed standards-based mathematical tasks that can vary in difficulty, context, and type. These tasks (aka working problems) are the context for connecting content and practices (which you’ll notice are connected in the sub-claims that we’ve already looked at).
In this activity, your group will focus on one of the 8 practice standards. (assign numbers for each of the groups to do) Within your “standard specific” group, you are charged with brainstorming a list of <*>What does the practice standard mean?, <*>How do students apply your practice standard to math content? and <*>How do you elicit this practice in a classroom? Each group will create a poster of your brainstorming ideas to be displayed on the wall under your particular standard. (Allow 10 minutes for brainstorming, poster creation. Circulate among groups, prepared with suggestions, etc.) Hand out the mathematics practice bullet points and have people update their posters.
Practices function differently from content in curriculum and instruction. <*>It is possible to teach content without practices, <*> but you cannot teach practices without content. <*>Practices should be authentically connected to specific content.
Within the Standards for Mathematical Content, there are three classes of standards, with respect to the practices: <*>Practice-integrated – content standards in which one or more practices is <*>explicit. Let’s look at an example of this practice in a content standard. <*> 3.NF.3b refers to equivalent fractions and states “explain why the fractions are equivalent” This content standard explicitly requires mathematical practice 3 within the wording. Here is another example. <*> F-BF.2 refers to writing sequences recursively and with an explicit formula and to using them to “model situations.” This explicitly refers to Mathematical Practice 4 (Model with mathematics)
THE RELATIONSHIP BETWEEN PRACTICE STANDARDS AND CONTENT STANDARDS Practice-related – content standards in which one or more practices is <*>implicit. For example: <*> 4.NBT.5 The last sentence says to “illustrate and explain the calculation by using equations, rectangular arrays, and/or area models” and implicitly requires students to employ several practices. Which practices do you think the student would use to solve a tasks related to this content standard? “ Make sense of problems and persevere in solving them” (MP.1), “ Reason abstractly and quantitatively” (MP.2), “ Look for and make use of structure” (MP.7) “ Look for and express regularity in repeated reasoning? (MP.8).
The three types of content standards are: Practice-Integrated where the practices are Explicitly defined in the standard Practice-Related where the practices are Implicitly contained in the standard (include multiple practice standards within the content standard) And the third group where content standards are not associated with a particular practice. PARCC tells us that practices will be involved in all assessment components – not just the performance-based assessment. In fact, the practices are, by nature, threaded throughout the content standards.
Here’s a statement to “tattoo on your forehead:” <*>The Mathematical Practice Standards will be well represented on the PARCC Assessment. To accomplish this, PARCC will incorporate Practice-Forward Tasks.
A practice-forward task is one that aligns to a practice-integrated or practice-related <*>content standard. In other words, a requirement of a practice-forward <*>task is that it is unlikely or impossible for a student to earn full credit on the task without engaging in the practice. Our next activity demonstrates some possible general practice-forward tasks that can be used when assessing mathematical practices with content standards. Turn to page __8__ in your manual. Let’s look at the first one together.
Think about a practice-forward task that could be used to collect evidence in this situation. What mathematical practice would the student need to use to receive full credit on this type of task? <*>I am focusing on the phrase “carefully defining variables.” Would you agree that a student would need to Attend to Precision (Mathematical Practice 6) in order to receive full credit for this task? We have given you a lengthy list of general practice-forward tasks. Working with your shoulder partner, examine the list and discuss the practice standard for which this task collects evidence. Some tasks may involve more than one practice standard. Remember you are looking for those standards that are explicit and/or implicitly referred. As you talk with each other, keep in mind which of the practices makes it possible to complete the task (and without which the student is unlikely to be successful). You will not have time to complete the entire sheet. Work through as many as you can to help you clarify which practice standards are required in these practice-forward tasks. When you’ve had a chance to make your “conjectures,” you can consult the answer key provided on page 10. (Allow 10-15 minutes for this activity, depending on the time remaining in the session.)
PARCC has used general statements like the ones in this activity when writing the specifications for these assessment tasks. And here are PARCC’s answers to some of the questions that everyone has about how the practices will be incorporated into the assessments. PARCC assures us that the Mathematical Practices will be assessed on all of the PARCC assessments.
PARCC has identified 3 key instructional shifts that need to occur in mathematics. The first of these is focus. The first statement in the Common Core State Standards for Mathematics is “Toward greater focus and coherence (*) .” So, what do we mean by “focus?” (*) When you focus on something, you give it your undivided attention. It’s like when my children were little and they used to take my face in their hands and say “Mommy, listen to me.” Focus hones us in on what we are doing. So, what does it mean when we look at curriculum? Can a few of you tell me what you think? Please pop up and answer. Allow time for participant answers. Answers should include the following: When we look at curriculum in the lower grades, it means exploring fewer topics but in greater depth. In the high school curriculum, focus means a concentration of skills and practice into a smaller number of underlying principles. Focus means that the Standards significantly narrow and deepen the scope of how time and energy is spent in the math classroom, so that students have sufficient time to think, practice and integrate new ideas into their growing knowledge structure. It also allows time for the kinds of rich classroom discussion and interaction that support the Standards for Mathematical Practice. What’s in and what’s out. Teaching less, learning more. I love the focus of the standards. Now, if we could just add 1 or 2 more things…Focus compromised is no longer focus at all. Focus is critical to ensure that students learn the most important content completely rather just a broad survey of content, “an inch deep, and a mile wide.” (*)
Focus is the first condition of faithfully implementing the Standards. Sub-claim A (*) reflects the fact that if students are not focusing and succeeding in the major work of their grade, then they are not succeeding at all and are not on tract to college and career readiness (*).
So, here is the task that is NOT a PARCC task: “I was driving down the road in my 4 wheeled canoe (*) and all 3 wheels fell off (*), how many flapjacks (*) does it take to tear the roof off a doghouse (*)?”? Ok, so this one was OBVIOUSLY not a PARCC task? Why? Because I’m not being “coherent.” None of these things are connected. Coherence is the 2nd key instructional shift in mathematics education. So what does coherence mean in the when we look at curricula? Would a few of you share your thoughts? Allow participants time to share. Answers should include the following: Coherence is about mathematical connections and building on what was taught previously. (*) Some connections are within a grade level where several topics knit together (like multiplication and areas models in 3rd grade). More often, though, connected standards reach across multiple grade levels with a progression of increasing knowledge, skill, or sophistication. Can’t approach each grade as an individual event-need to see the progression. It’s knowing what comes before us and after us…knowing where to start and when to stop! Standards are not so much assembled out of topics, but woven out of progressions.
The 3rd major instructional shift for mathematics is “rigor.” Rigor includes three major components: Conceptual understanding Application Fluency We begin by building a conceptual understanding. Then, we have our students apply that knowledge over and over and in different and new situations. Over time, they can develop fluency. How do we build “rigor” into the curriculum? Answers could include the following: According to these standards, it is not enough for students to learn procedures by rote. Nor, on the other hand, is it enough for students to “understand the concepts” without being able to apply them to solve problems. Nor, finally, is it enough for students to learn the important procedures of mathematics without attaining skill and fluency in them. Conceptual understanding: Teachers teach more than “how to get the answer” and support students’ ability to access concepts from a number of perspectives so that students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of core math concepts by solving short conceptual problems, applying math in new situations, and speaking about their understanding. Application: Students are expected to use math and choose the appropriate concept for application even when they are not prompted to do so. Teachers provide opportunities at all grade levels for students to apply math concepts in “real world” situations. Teachers in content areas outside of math, particularly science, ensure that students are using math – at all grade levels – to make meaning of and access content. Procedural skill and fluency. (See next set of slides.)
Let’s talk a little more about fluency. Within some of the major clusters, you will find standards that use the word “fluently.” Whenever “fluently ” appears in a content standard, it means “quickly and accurately,” much the same as it means to be fluent in a foreign language. (*) Fluent isn’t halting, or stumbling, or reversing oneself. To be fluent is to flow. Fluency in this sense isn’t something that happens in a single grade level. Before fluency is expected at a particular grade level, the roots for understanding and the opportunities for sufficient practice and additional support are provided at one or more of the earlier grade levels. (*) Share a personal story (or make this one your own): My son was watching the Olympic trials the other night. I was in the kitchen cooking supper, and he said to me, “Wow, they are running really fast to be running for such a long time.” I asked, “How far did they run? How long did it take?” He replied that it was the 10,000 meter race, it took them about 32 minutes, and their average lap speed was about 70 seconds. And he asked, “So, how many miles is that?” So my first thought was to start with the 32 minutes. I decided to think of the average speed as a little over a minute, so they ran around the track about 32 times. I remembered in high school that it took 4 laps to make a mile, so I said, well it has to be less than 8 miles, but I’m leaving about 10 seconds a lap unaccounted for, so that’s 320 seconds about 5 laps so 1.25 miles, I’ll make up for those extra seconds, so that means it’s around 6.75 miles. So, I went back to cooking, and I was standing there thinking about it some more, and I thought to myself, “Well, I know 1 inch is 2.54 centimeters (*)---well, that’s way too much math to do in my head-that will have to wait until later. So what else do I know?” Then I thought, “Well, I know that 3.1 miles is a 5K…and then it hit me, well, duh…10,000 meters is a 10K, so it must be about 6.2 miles.” So…am I fluent in converting within the metric system…not really, because if I were fluent, those synapses in my brain would have immediately connected 10,000 meters to a 10K then I could have gone directly to 6.2 miles. Did I incorporated good mathematical practices while working this problem? Can someone name one and tell me how that I used it working this problem? Sample answers could include: MP. 1: Making sense of problems and persevere in solving them. (looked for entry points into the problem, monitor and evaluate progress and changed course) MP. 2: Reason abstractly and quantitatively. (considered the units involved) MP. 3: Construct viable arguments and critique the reasoning of others. (made conjectures and built a logical progression to explore the truth of the conjecture) MP. 4: Model with mathematics (applied what I comfortable making assumptions about and approximated to simplify a complicated situation, realizing it might need revision later) How do we become fluent? popcorn answers; sample answers include: So, fluency is definitely developed using mathematical practices over and over. In fact, fluency requires practice of both content and practices. Fluency isn’t something that happens in a single grade level. There has to be a depth of understanding at the conceptual level and opportunities for sufficient practice. It is not about rote memorization but about knowing and understanding conceptually. I hope today’s two sessions have built a foundation of understanding as we move becoming fluent with the standards. (*)
Each grade level K-8 of the Model Content Frameworks indicates the Fluency Expectations (*). For grade 3-6, fluency is sub-claim E (*) of the Math Master Claim. While the PARCC assessments will not address fluency at the high school level, fluency is still important, and the Model Content Frameworks make Fluency Recommendations (*) . In the same section of the PARCC Model Content Frameworks, “culminating standards” (*) are discussed. Culminating standards represent the “end” of a series of standards. For example for operations with rational numbers, after we have added, subtracted, and multiplied, the culminating standard is “divide.” More here on culminating standards??????
Now for the summary slide that breaks all the PowerPoint rules…. This morning in the plenary session, we started with the Mathematics Master Claim(*): Students are on-track for college and career readiness. In the math sessions today, we connected the PARCC claims to both the Standards for Mathematical Content (*) and the Standards for Mathematical Practice (*). In session 1, we looked at the practices which are included in the sub-claim A (*), sub-claim B (*), sub-claim C (*), and sub-claim D (*). Then, in this session, we looked at the classifications of the standards as major/additional/supporting. Sub-claim A refers to major content (*)-some of these standards in grades 3-6 require fluency which is sub-claim E (*) and we classified some standards as additional and supporting content (*) which is included in sub-claim D (*). To successfully implement the CCSS we all need to narrow our FOCUS. We also need to be able to see the COHERENCE between the standards. So we hope this session will help you focus on the standards and be able to communicate them coherently in your states. Thank you again for your attention and participation.
