Implementing Math

1
Implementing Math Science Initiatives at the
Federal State Levels What Needs to Happen to
Move Forward

• Hank Kepner
• National Council of Teachers of Mathematics,
• Past-President
• University of Wisconsin-Milwaukee
• Milwaukee Public Schools
• kepner_at_uwm.edu

2
Common Core State Standards for Mathematics

• Standards for Mathematical Practice
• K8 Grade level standards
• Domains across grade levels
• High School standards
• conceptual categories

3
Features of the Common Core

• Focus attention on core concepts in number and
  numeration their relationships to operations
  with a focus on the structure of the number
  system
• Aggressive timelines for teaching particular
  concepts in elementary and middle grades
• They introduce multiple measurement systems
  (metric, non-standard English) simultaneously
  tie the number line directly to scales to improve
  students visualization of number relationships
• They support the articulation of some key
  learning trajectories in numeration and geometry
  and
• They remain agnostic about sequencing and
  organization of high school math.

4
Standards for Mathematical Practice

• Mathematically proficient students
• Make sense of problems and persevere in solving
  them
• Reason abstractly and quantitatively
• Construct viable arguments and critique the
  reasoning of others
• These are measures of Student behavior!

5
Standards for Mathematical Practice

• Mathematically proficient students
• 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
• These are measures of Student behavior!

6
The MSP programs can play a critical role in
maximizing this opportunity

• To strengthen the movement towards Common Core
  Standards implementation
• To get your innovations to move to scale to the
  degree to which they can tie into a broader
  policy agenda

7
Window of Opportunityfrom now until 2014- 2015
when assessments are put into place.

• The current circumstance is unstable and
  ambiguous at the state level.
• The standards will make sense only when we have
  instructional and assessment exemplars to use and
  analyzethe operational definitions!
• Most standards do not describe depth of cognitive
  demand to be assessed. Caution about trivial
  level
• Monitor and influence the assessment developments
  to ensure sound assessment beyond multiple
  choice
• Partnership for the Assessment of Readiness for
  College and Careers (PARCC includes Achieve)
• SMARTER Balanced Assessment Consortium

8
Recent challenges to Mathematics for All
students and constraints on STEMCurrent state
sanctions and district policies have led
administrators to emphasize only that mathematics
content which is assessed and to de-emphasize or
ignore what is not assessed.

9

• The result in K-8 mathematics has been a
  narrowing of delivered curricula, a test-driven
  curriculum, in each state that
• Has clarity and specificity (expectations defined
  with sufficient detail to communicate intent and
  applicability)
• Lacks coherence (expectations for mathematics
  content and processes are marked by logical
  disconnections and inappropriate trajectories)
• Lacks focus (insufficient time available to learn
  concepts and skills critical for understanding
  the expected content, in part due to re-teaching
  of (un)learned content).
• Campbell

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Simply teaching more mathematics/science content
to teachers is not the answer!

• Teachers not only need to know the content they
  teach (including notation, language and
  definitions),

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• Teachers also need to know what
  mathematics/science to access and use it when
  they
• Pose math/science questions,
• Evaluate support students explanations,
• Use or choose ways of representing the
  math/science,
• Choose, sequence design tasks examples,
• Determine whether and how to provide
  explanations,
• Analyze/address student misconceptions,
  errors, and

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Address Students Lack of Engagement

• Lack of engagement in learning is the greatest
  problem in math classrooms
• How can the CCSS help us with this?
• The Mathematical Practices as a way to leverage
  discourse
• Formative Assessment or Assessment for Learning
• New technologies for networking

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Goal for STEM education

• Schooling is about modeling our world
  (scientifically, technologically, mathematically,
  and statistically),
• feeling engaged and empowered,
• building opportunities for expressiveness
• fostering collaboration, designing and testing
  solutions,
• encouraging active citizenry, and
• being well-prepared to earn a satisfying and
  sufficient living.

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Implementation of the Standards for Mathematical
PracticeStandard 3Construct viable arguments
critique the reasoning of others.
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Construct viable Arguments Critique the
reasoning of others

• understand use stated assumptions, definitions,
  and previously established results in
  constructing arguments
• make conjectures and build a logical progression
  of statements to explore the truth of their
  conjectures
• justify their conclusions communicate them to
  others
• listen to or read the arguments of others, decide
  whether they make sense, and ask useful questions
  to clarify or improve the arguments.

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Length Number of Beams
2 3 5 10
50 n
17
3 4(n-1)
n 2n (n-1)
3n (n-1)
4(n-1) 3
4n - 1
Next Now 4 Start 3
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Construct viable Arguments Critique the
reasoning of others

• understand use stated assumptions, definitions,
  and previously established results in
  constructing arguments
• make conjectures and build a logical progression
  of statements to explore the truth of their
  conjectures
• justify their conclusions communicate them to
  others
• listen to or read the arguments of others, decide
  whether they make sense, and ask useful questions
  to clarify or improve the arguments.

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A Challenge of Phasing in Implementation
Year Grade Level
2010-11 K-1
2011-12 2-3
2012-13 4-5
2013-14 6-7
2014-15 8
A critical area that must be addressed in state
implementation plans. -- Include the
mathematical practices in all aspects of
implementation.
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Phasing the Implementation

• Simultaneously
• new curricular development including tasks
  addressing Standards for Mathematical Practice,
• build professional development systems that
  expand instructional strategies,
• comprehensive assessment systems,
• build technological infrastructures

21
The need to couple curriculum, classroom
instruction, assessment via the CCSS
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A learning trajectory/progression is

• a researcher-conjectured, empirically-supported
  description of the ordered network of constructs
  a student encounters through instruction (i.e.
  activities, tasks, tools, forms of interaction
  and methods of evaluation), in order to move from
  informal ideas, through successive refinements of
  representation, articulation, and reflection,
  towards increasingly complex concepts over time
• (Confrey et al., 2009)

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Standards Progressions

• The Math Common Core presents and builds on
  conventional wisdom and experiences from best
  practices in presenting empirically-supported
  description of the ordered network of constructs
  a student encounters through instruction
  (activities, tasks, tools, forms of interaction
  and methods of evaluation), in order to move from
  informal ideas, through successive refinements of
  representation, articulation, and reflection,
  towards increasingly complex concepts over time
  Confrey et al., 2009

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Common Core Fractions Number Operations - Gr 3

• 3.NF Develop an understanding of fractions as
  numbers.
• 1. 1/b as quantity when whole is partitioned
  into b equal parts.
• 2. represent a/b on number line
• 3. a/b mark off a lengths 1/b from 0.
  Endpoint is number a/b - cardinality!
• 4. Equivalences (same size, same point)

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Common Core Fractions Number Operations Gr 4

• 4.NF Extend understanding of fraction
  equivalence and ordering.
• Build fractions from unit fractions, extend
  understanding of whole number operations.
• Understand decimal notation for fractions,
  compare decimal fractions.
• Gr 4 Limited domain-denominators
  2,4,5,6,8,10,12,100

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Common Core Fractions Number Operations Gr 5

• 5.NF
• 1. Use equivalent fractions as strategy to
  add/subtract fractions.
• 2. Apply extend understanding of mult/div to
  multiply divide fractions.

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Assessment

• The elephants in the room.
• Accountability high-stakes testing of students
• and teachers.
• Assessment for learning.
• Formative assessment.
• Diagnostic assessmentRtI, other

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Assessment for Learning

• The UK Assessment Reform Group (1999) identifies
  FIVE PRINCIPLES OF ASSESSMENT for LEARNING
• The provision of effective feedback to students.
• The active involvement of students in their own
  learning.
• Adjusting teaching to take account of the results
  of assessment.
• Recognition of the profound influence assessment
  has on the motivation and self esteem of pupils,
  both of which are critical influences on
  learning.
• The need for students to be able to assess
  themselves and understand how to improve.

Assessment for Learning Defined Stiggins et al.
(2005) http//www.assessmentinst.com/wp-content/u
ploads/2009/05/afldefined.pdf
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Formative Assessment

• a process used by teachers and students during
  instruction that provides feedback to adjust
  ongoing teaching and learning to improve
  students achievement of intended instructional
  outcomes.
• rely on learning progressions (Heritage 2008)
• share explicit learning goals
• provide students with descriptive feedback
• promote a collaborative environment
• include peer and self assessments
• (CCSSO, 2008)

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Defining Diagnostic Assessment

• Diagnostic Assessment requires an explicit theory
  of how a students thinking progresses over time.
  
• Comes from a combination of dia, to split apart,
  and gnosi, to learn, or knowledge
• Efficient, effective use of students responses,
  both to document growth in their learning
  understanding (content and process) and to
  promote that growth, from initial states to more
  powerful, coherent and aligned conceptions

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Issues in Diagnostic Assessment

• Focus on big ideas in mathematics
• Focus on how childrens mathematics understanding
  evolves over time, rather than replicating the
  structure of mathematics
• Rapid feedback to teachers on individual and
  group status of understanding
• Provide recommendations and interventions for
  improving growth, remedying gaps in conceptual
  understanding
• Professional Development continuous
  instructional improvement

32
Defining and deploying a broader
college-and-career STEM agenda of a states
standards that builds on the CCSS.
33
Next Steps for the Common Core Standards

• The high school standards vary widely in grain
  sizethere is substantive need to unpack many of
  the standards to clarify how the sub-constructs
  develop and build on each other.
• There is NOT a single sequence.
• The document is weak in relation to career in
  the promise of College and Career Ready
  Standards
• I would argue the developers lost focus on this!

34
Appendix AHigh School Course Pathways

• These are examples -- NOT mandated paths!
• 4 possible approaches to organizing the content
  across several courses - by no means an
  exhaustive list.
• States, curriculum developers, schools may use as
  guides in preparing instructional materials and
  assessments (End of course?)

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High School Course Pathways

• Appendix A provides samples for
• Algebra-Geometry-Algebra 2 a sequence used only
  in the US, and
• Integrated possibilities consider IB,
  international models, several US models.
• Shaughnessy, J.M. (2011) An Opportune Time to
  Consider Integrated Mathematics. www.nctm.org/pre
  smess0311.

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Serious STEM Omissions

• The math CCSS are conservative, and delayed on
  modeling, probability and statistics, and rate of
  change and early functions as an introduction to
  algebra.
• Further, they only modestly focus on the use of
  new learning technologies this could leave our
  students foundering in exciting arenas
  incorporating visualization, integration of
  topics, engineering, and design and the use of
  simulations. These topics constitute a critical
  agenda for our field, and we need a strategy to
  avoid their marginalization.

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Mathematics for Non-STEM

• Often cited critical areas
• Confidence, communication, reasoning
• Emphasis on ratio, proportional reasoning, and
  interpretations
• Rate of change Modeling
• Discrete Probability, Data Analysis/Statistics
• Collect, organize, analyze, interpret in context,
  inferences
• Discrete Math topics
• Strategic use of technology
• Calculators, spreadsheets, dynamic geometry, CAS
• Why arent these for STEM students, too?

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High School Course Pathways

• A question of my science colleagues
• Will a high school science sequence and courses
  be aggressively studied in terms of STEM
  integration needs or will it remain in
  disciplinary silos?

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ELA standards Reading in Science and Technical
Subjects

• Key ideas and details
• Follow precisely a multistep procedure when
  carrying out experiments, taking measurements, or
  performing technical tasks.
• Reading Standards for Literacy in Science and
  Technical Subjects
• Craft and Structure
• Determine the meaning of symbols, key terms, and
  other domain-specific words and phrases as they
  are used in a specific scientific or technical
  context relevant to grades 68 texts and topics.
• Integration of Knowledge and Ideas
• Compare and contrast the information gained from
  experiments, simulations, video, or multimedia
  sources with that gained from reading a text on
  the same topic (grades 6-8)
• Range of Reading and Level of Text Complexity

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Writing Standards for Science and Technical
Subjects

• Text Type and Purposes
• -- Write arguments focused on
  discipline-specific content
• Production and Distribution of Writing
• Develop and strengthen writing as needed by
  planning, revising, editing, rewriting, or trying
  a new approach, focusing on addressing what is
  most significant for a specific purpose and
  audience
• Research to Build and Present Knowledge
• Conduct short as well as more sustained research
  projects to answer a question (including a
  self-generated question) or solve a problem
  narrow or broaden the inquiry when appropriate
  synthesize multiple sources on the subject,
  demonstrating understanding of the subject under
  investigation

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Developing advanced systems for professional
development around points 1-3

• Professional development needs to be
• Content and curricular specific
• Closely linked to classroom practices
• Include intensive learning opportunities with
  year round follow up
• Obligatory
• Linked to building professional communities
• States need professional development systems
  connecting district resources, departments of
  public instruction, universities, colleges,
  community colleges.

42
Developing advanced systems for professional
development around points 1-3

• PD tied to standards -- including their
  interpretation and connections.
• The Standards of Mathematical Practice are
  STANDARDS that the participating states have
  signed on to implement.
• CAUTION Too many implementation and assessment
  programs are already seeking to ignore or avoid
  the Standards for Mathematical Practice.
• Placing attention and focus only on content
  standards is insufficient!

43
Using longitudinal data systems to decipher and
study curricular effectiveness and provide
empirical support for change.
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Priority Six Using longitudinal data systems to
decipher and study curricular effectiveness and
provide empirical support for change.

• On Evaluating
• Curricular
• Effectiveness
• (NRC 2004)

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Recent Advances

• Implementation fidelity (Huntley, 2010)
• Opportunity to learn and related constructs
  (McNaught et al. 2010)
• Fair Tests for comparison (Chavez et al. 2010)
• Complexity of relationships among teacher and
  student variables and curricular effects Tarr et
  al. (http//cosmic.missouri.edu/aera10/)
• We need to continue to build on these to ensure
  that we can substantiate future effectiveness.

46
NEXT STEPSandThe Care and Feeding of the
Standards
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Next Steps for the Common Core Standards

• The Mathematical Practice Standards are presented
  independent of content standardsrisk of being
  isolated or ignored
• Use of and competency with technology is not
  adequately or constructively addressed for
  students
• There is limited attention to engaging students
  in mathematics through modeling with a robust and
  intriguing use of technology and skills required
• Yet to be done Math connectionsacross domains
  and interdisciplinary concepts applications
• The wording of many standards is obtusecombining
  math propositions with combination of content and
  pedagogical advice, verbs hard to interpret
  for assessment

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Care and Feeding of the Standards

• Should include
• members of professional organizations
• teachers
• researchers,
• assessment experts
• Should have a three part timeline immediate
  fixes, minor revisions and major reviews
• Should be an NRC type process with documented
  responses to feedback

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Sample Decisions that Should Be Re-examined over
time

• The practices of mathematics are presented
  independently from the content standards they
  could be isolated, under-emphasized, and without
  careful professional development
• There is limited detail to engaging students in
  mathematics through modeling with robust use of
  technology, and developing the skills and
  concepts of modeling
• The important need for mathematical connections
• across mathematical domains and concepts
• interdisciplinary fields, concepts and
  applications
• is mentioned, but there is a lack of guidance or
  specific examples

50
Sample Decisions that Should Be Re-examined over
time

• New standards for Probability and Statistics in
  early grades should supplant the current weak
  treatment of the topic in Measurement
• The CCSS construe Number narrowly tend to
  overemphasize additive structureslimiting early
  and foundational development of
  multiplicative/divisional structures related to
  ratio and rate, and many algebraic patterns of
  growth

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Major CCSS Focus

• The success of the standards depends on the
  ability of teachers to assist students in
  learning the specified fewer standards at grade
  level.
• Therefore, the success of the standards should be
  measured heavily, though not exclusively, on the
  narrowing of the performance gaps

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Major Concern NOT Addressed

• The CCSS for Mathematics present a lock-step
  sequence of content for grades K- 8.
• For high performing students, how will this be
  addressed in typical school implementation and
  accountability assessments?

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Conclusions

• The Common Core State Standards present
  opportunities for innovative thinking around
  students engagement in curriculum
• Require a plan for phasing in CCSS by grades and
  domains
• Need to elaborate the learning trajectories
• Need to inter-relate curriculum, instruction, and
  assessment using new technologies as a priority
  to improve engagement and fair
• Continue evaluating curricular effectiveness

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Your Opportunities for Reflection
Thanks, Hank

• Thanks to the work of an entire field of those
  who are tirelessly seeking to implement the CCSS.
• Campbell, P. (2011) Addressing Challenges in the
  Common Core Mathematics Specialists in
  Elementary and Middle Schools. mathedck.files.word
  press.com/2011/02/jointmath2011pfcrev.ppt
• Confrey, J. (2011) Implementing Math Science
  Initiatives at the Federal State Levels What
  Needs to Happen to Move Forward
• NCTM. (2011) Common Core Standards Implementation
  http//www.nctm.org/ (NCTM posts and pointers
  to sources)
• Shaughnessy, J.M. (2011) An Opportune Time to
  Consider Integrated Mathematics. www.nctm.org/pres
  mess0311.
• Hank Kepner
• kepner_at_uwm.edu