3. Construct viable arguments and critique the reasoning of others.Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
Distribute the Bell Ringer –Nested Circles and Squares and ask the participants to work individually. Participants can use the Pythagorean theorem, the relationship of the hypotenuse to a leg in a 45-45-90 right triangle, or trigonometry- sin 45 degrees. Use chart paper for a group to draw the picture of the nested circle and square.
Discuss definition briefly.
The important point here is lesson planning from the students’ point of view. The mathematical practices are “student” mathematical practices.
Here is an overview for the content in Grades 3 -8
How do we address the (+) standards? What does the star (*) mean? Why is Modeling a conceptual category?
Let’s look at an example of Focus using Number and Operations from K to high school. These are progressions that are necessary for student to master to be proficient to move to the high school.
Leading the Teaching and Learningof Mathematics in the CCSS Era! As you enter the room At your tables…. Choose a corner and list 2-3 vital teacher team behaviors essential to highly effectiveInstructional practices used in your school or by your team…Do not write in the middle circle of the sheet! Thank you! Dr. Timothy Kanold tkanold.blogspot.com Dr. Timothy Kanold 2012 tkanold.blogspot.com
Our outcomes for this session1) Examine your history as a PLCcollaborative team2)Examine Mathematics Teaching andLearning through the lens of the CCSSMathematical Practices2) Discuss Lesson Planning Tools toimplement the Standards forMathematical Practice Dr. Timothy Kanold 2012 tkanold.blogspot.com
Common Core Mathematics in aPLC at Work™ Series Ch. 2
Ch. 1: The Paradigm ofCollaborationUsing High PerformingCollaborative Teamsfor Mathematics… Better at … Dr. Timothy Kanold 2012 tkanold.blogspot.com
Leading the Teaching and Learningof Mathematics in the CCSS Era! Complete the Team History tool p.1 -2 Dr. Timothy Kanold 2012 tkanold.blogspot.com
Our 2nd outcome for thissessionExamine Mathematics Teaching andLearning through the CCSSMathematical Practices Dr. Timothy Kanold 2012 tkanold.blogspot.com
http://www.flickr.com /photos/shawnparker photo/6637823915/in /photostream/ How students learn... and demonstrate proficiency in Mathematics…Dr. Timothy Kanold 2012 tkanold.blogspot.com
Noel Tichy – Your TeachablePoint of View or TPOV“A cohesive set of ideas and concepts that a person is able to clearly articulate to others.” Director, Global Leadership Program & Professor of Management and Organizations tkanold.blogspot.com Dr. Timothy Kanold 2012
Your TPOV for Effective InstructionFind your poster paper on the Wall… work with your colleagues to create a “Matchbook” description of your vision for effective instruction…18 words or less – pictures allowed Dr. Timothy Kanold 2012 tkanold.blogspot.com
THE CCSS TPOV for MathematicsInstructionUnit and Lesson design will require a depth of conceptual understanding andprocedural fluency regardless of the content…demonstrated by the students. Dr. Timothy Kanold 2012 tkanold.blogspot.com
Learning how and why is now part of the guaranteed and viable curriculum!Common Core State Standards U N D E RMathematical S Mathematical T Content A Practices N D I N G Dr. Timothy Kanold 2012 tkanold.blogspot.com
THE CCSS TPOV for MathematicsInstruction HO p.3Built on a foundation of the 8 standards for Mathematical Practice Dr. Timothy Kanold 2012 tkanold.blogspot.com
The Standards for MathematicalPractice (See Handout p.3-7)Choose Practice 1,2, 3, 4, 5 or 6Highlight the verbs that illustratestudent actions!Circle, highlight or underlinephrases for your chosenpractice… Dr. Timothy Kanold 2012 tkanold.blogspot.com
The Common Core Standards for[Student] Mathematical Practice p.291. What is the intent and why it isimportant?2. What teacher actions help todevelop this CCSS MP?3. What evidence exists that studentsare demonstrating this MP?MP1: Explain and Make Conjectures…Book p. 31-32 Dr. Timothy Kanold 2012 tkanold.blogspot.com
Developing Reasoning Habits of Mind1)Provide tasks that require students to figure thingsout for themselves (The AHA moment)2)Move from Empirical (experiment that supportssome cases), to pre-formal (intuitive) to formal(arguments for mathematical certainty)3)Plan for and expect student communication oftheir reasoning to classmates and the teacher –using proper vocabulary4)Use questions and prompts such as “How do youknow?” And “Why does this work?” Dr. Timothy Kanold 2012 tkanold.blogspot.com
The Standard for MathematicalPractice #3 Book p. 37MP # 3 Construct viable arguments andcritique the reasoning of others1) Students make conjectures 2) Students justify their conclusions and communicate them to others 3) Students compare the effectiveness of two plausible arguments 4) Students listen and respond to the arguments of others for sense making and clarity Dr. Timothy Kanold 2012 tkanold.blogspot.com
3. Construct viable arguments and critique the reasoning of othersDr. Timothy Kanold 2012 tkanold.blogspot.com
The Standards for MathematicalPractice HO p.8Rich Mathematical Tasks… Qualitative Reasoning and McDonald’sWith a shoulder partner!Wikipedia reports that 8% of all Americans eat at McDonalds every day310 Million Americans and 12,800 McDonalds…Make a conjecture and create a mathematical argument…
The Standards for [Student]Mathematical PracticeSMP # 4 Model with MathematicsMathematically proficient studentscan apply the mathematics theyknow to solve problems arising in everyday life, society, and the workplace… Dr. Timothy Kanold 2012 tkanold.blogspot.com
The Standards for [Student]Mathematical Practice SMP # 2 Reason abstractly and quantitativelyMathematically proficient studentsmake sense of quantities and their relationships in problem situations… Contextualize and De-contextualize Dr. Timothy Kanold 2012 tkanold.blogspot.com
Unit by Unit planning for highcognitive demand tasks…N.Q.1: Use units as a way tounderstand problems and to guide thesolution of multi-step problems;choose and interpret units consistentlyin formulas; N.Q.3: Choose a level of accuracyappropriate to limitations onmeasurement when reportingquantities. Dr. Timothy Kanold 2012 tkanold.blogspot.com
The Power of a Stage 6 and 7Collaborative Team…) HO p. 8In your teacher Teams:Discuss your expectations for studentdemonstration of quality work in defense oftheir mathematical argument for theproblem.Discuss how your lesson plan for thisproblem would promote studentcommunication of their argument withothers and respond to one another basedon their solution defense. Dr. Timothy Kanold 2012 tkanold.blogspot.com
The Standards for [Student]Mathematical Practice HO Page 9
Shoulder Partner Discussion…•To what degree do you believe yourstudents are currently demonstratingproficiency in the standards formathematical practice?• How might you use this information so farto identify starting points for your workwith the Standards for MathematicalPractice? Dr. Timothy Kanold 2012 tkanold.blogspot.com
Our 3rd outcome for thissession3) Discuss Lesson Planning Tools tohelp you implement the Standardsfor Mathematical Practice Dr. Timothy Kanold 2012 tkanold.blogspot.com
Planning Lessons Together!• Professional Learning Communities are essential to good planning.• Read the Elements of Effective Lesson Design (HO p.10-11)Or book p.46-49 Explanations – page 49-57• How are each of these elements connected to your current teaching? Dr. Timothy Kanold 2012 tkanold.blogspot.com
CCSS Mathematical PracticesLesson Design ToolTake a moment to scan the elementsof this lesson design and/or reflectiontool…How could you use this tool with yourteam in 2012-2013? Dr. Timothy Kanold 2012 tkanold.blogspot.com
Our outcomes forthe contentsession1) Examine the difference betweenrelevant and meaningful mathematics2)Examine parts of the CCSS HighSchool content and 6-12 progressions3) Discuss Course Scope andSequencing for grades 6-12 Dr. Timothy Kanold 2012 tkanold.blogspot.com
Common Core Mathematics in aPLC at Work™ Series Ch.3
The Mathematics Curriculumof the CCSS…With a shoulder partner… – Share your understanding of the difference between relevant mathematics and meaningful mathematics Dr. Timothy Kanold 2012 tkanold.blogspot.com
Relevance Vs. MeaningRelevant mathematics:References the context for the lesson as part of essential mathematics and mathematical tasks the student needs to know. Ask yourself….Does the lesson present important and essential mathematics? Dr. Timothy Kanold 2012 tkanold.blogspot.com
Relevance Vs. MeaningMeaningful mathematics:References the context for the lesson as containing elements that create meaning, reasoning and sense making for the student - while also connecting to the students’ prior knowledge and understanding… Dr. Timothy Kanold 2012 tkanold.blogspot.com
Our 2nd Outcome for thissessionExamine parts of the CCSS High Schoolcontent and 6-12 progressions Dr. Timothy Kanold 2012 tkanold.blogspot.com
Standards for MathematicalContent (book p. 66) Dr. Timothy Kanold 2012 tkanold.blogspot.com
High School Conceptual CategoriesConceptual categories in high school (App. C p. 175)• Number and Quantity• Algebra• Functions• Geometry• Statistics and ProbabilityCollege and career readiness threshold• (+) standards indicate material beyond the threshold or needed for advanced courses; can be in courses intended for all students.• (*) specific modeling standards Dr. Timothy Kanold 2012 tkanold.blogspot.com
Standards for Mathematical Content Conceptual Categories • Domains are larger groups of related standards. • Clusters are groups of related standards. • Standards define what students should understand and be able to do during a unit… Dr. Timothy Kanold 2012 tkanold.blogspot.com
HS: Conceptual Category - Geometry The 6 Domains – Congruence – Similarity, Right Triangles, and Trigonometry – Circles – Expressing Geometric Properties with Equations – Geometric Measurement and Dimension – Modeling with Geometry Dr. Timothy Kanold 2012 tkanold.blogspot.com
Conceptual Category– GeometrySamplebook p. 183-187• List the domainsforGeometry on chart paper - horizontally• List the clusters of standards for each domain vertically…and count the number of standards in each cluster (how many are college prep and how many are advanced?)Which clusters/standards appear to be new or more challenging for each of the domains? Dr. Timothy Kanold 2012 tkanold.blogspot.com
Geometry Progressions…Middle school foundations• Hands-on experience with transformations.• Low tech (transparencies) or high tech (dynamic geometry software).High school rigor and applications• Properties of rotations, reflections, translations, and dilations are assumed, proofs start from there.• Connections with algebra and modeling Dr. Timothy Kanold 2012 tkanold.blogspot.com
7-12 Increased emphasis…• Statistics and Probability – Interpreting Categorical and Quantitative Data – Making inferences and Justifying Conclusions – Conditional Probability and the Rules of Probability – Using Probability to Make Decisions Dr. Timothy Kanold 2012 tkanold.blogspot.com
Discuss at your tablesWhat needs to be done in your district, school or department to look at the conceptual categories, clusters, standards, and progressions in the 6-8 or the high school curriculum so that all teachers understand…The expectations of the grade level content? Dr. Timothy Kanold 2012 tkanold.blogspot.com
Our 3rd Outcome for thissessionResources… Dr. Timothy Kanold 2012 tkanold.blogspot.com
www.mathccc.org Dr. Timothy Kanold 2012 tkanold.blogspot.com
Mathematics Assessment Project(MAP)http://map.mathshell.org.uk/materials Dr. Timothy Kanold 2012 tkanold.blogspot.com
Tools for the Common Core Standardscommoncoretools.wordpress.com Dr. Timothy Kanold 2012 tkanold.blogspot.com
Mathematics Assessment Project(MAP)http://map.mathshell.org.uk/materials• 20 ready-to-use Lesson Units for Formative Assessment for high school. cross referenced to CCSS content and practices standards. (Ultimately 20 per grade 7-12)• Summative assessments, aimed at “College- and Career-Readiness,” presented in two forms: (1) a Task Collection with each task cross- referenced to the CCSS, and (2) a set of Prototype Test Forms showing how the tasks might be assembled into balanced assessments. Dr. Timothy Kanold 2012 tkanold.blogspot.com
The Illustrative Mathematics Projectillustrativemathematics.org• Hyperlinked CCSS• Developing a complete set of tasks for each standard – Range of difficulty – Simple illustrations of single standards to complex tasks spanning many standards.• Provide a process for submitting, discussing, reviewing, and publishing tasks. Dr. Timothy Kanold 2012 tkanold.blogspot.com
illuminations.nctm.org/Lessons.aspx Dr. Timothy Kanold 2012 tkanold.blogspot.com
End of Day Reflections1. Are there any aspects of your own thinking and/or practice that our work today has caused you to consider or reconsider? Explain.2. Are there any aspects of your students’ mathematical learning that our work today has caused you to consider or reconsider? Explain.3. What would you like more information about?