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Explore geometry learning progressions, plan lessons, anticipate student responses, assess understanding, and reflect on teaching strategies. Enhance math instruction through deductive reasoning and geometric experiences.
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CCRS Quarterly Meeting # 3Unpacking the Learning Progressions http://alex.state.al.us/ccrs/
Alabama Quality Teaching Standards 1.4 2.7 5.3 1.4-Designs instructional activities based on state content standards 2.7-Creates learning activities that optimizeeach individual’s growth and achievement within a supportive environment 5.3-Participates as a teacher leaderand professional learning community member to advance school improvement initiatives
The Five Absolutes + A Balanced Instructional Core = A Prepared Graduate Five Absolutes The Instructional Core • Teach to the standards (Alabama College- and Career-Ready Standards – Math Course of Study) • Aclearly articulated and “locally” aligned K-12 curriculum • Aligned resources, support, and professional development • Regular formative, interim/benchmark assessments to inform the effectiveness of the instruction and continued learning needs of individual and groups of students • Each student graduates from high school with the knowledge and skills to succeed in post-high school education and the workforce 1.4 2.7 5.3
Outcomes Participants will: • Reflect on Next Steps from QM #2 • Review and deepen understanding of the Geometry Learning Progression and how the content is sequenced within and across the grades (coherence) • Examine specific mathematical content along “Standards Progressions” to uncover necessary shifts in when topics are addressed and mastered. • Illustrate, using tasks, how math content develops over time and discuss how the progressions in the standards can be used to inform planning, teaching, and learning
Next Steps Identify standards and select a high level task. Plan a lesson with colleagues. Anticipate student responses, errors, and misconceptions. Write assessing and advancing questions related to student responses. Keep copies of planning notes. Teach the lesson. When you are in the Explore phase of the lesson, tape your questions and the students responses, or ask a colleague to scribe them. Following the lesson, reflect on the kinds of assessing and advancing questions you asked and how they supported students to learn the mathematics.
CCRS-Mathematics Learning Progressions
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Analysis of Deductive Systems Geometric experiences influence advancement through the levels Deductive Systems of Properties Relationships Among Properties 5. Rigor Properties of Shapes 4.Deduction Classes of Shapes 3. Informal Deduction Instruction at a higher level than that of the student likely results in rote learning with little understanding Shapes 2. Analysis 1. Visualization Van Hiele Levels of Geometric Reasoning
Four Characteristics: The Van Hiele levels of geometric reasoning are sequential. Students must pass through all prior levels to arrive at any specific level. These levels are not age-dependent in the way Piaget described development. Geometric experiences have the greatest influence on advancement through the levels. Instruction and language at a level higher than the level of the student may inhibit learning.
1. VisualizationStudents can name and recognize shapes by their appearance, but cannot specifically identify properties of shapes. Although they may be able to recognize characteristics, they do not use them for recognition and sorting. Suggestions for instruction using visualization: sorting, identifying, and describing shapes manipulating physical models seeing different sizes and orientations of the same shape as to distinguish characteristics of a shape and the features that are not relevant building, drawing, making, putting together, and taking apart shapes.
2. AnalysisStudents begin to identify properties of shapes and learn to use appropriate vocabulary related to properties, but do not make connections between different shapes and their properties. Irrelevant features, such as size or orientation, become less important, as students are able to focus on all shapes within a class. They are able to think about what properties make a rectangle. Students at this level are able to begin to talk about the relationship between shapes and their properties. Suggestions for instruction using analysis shifting from simple identification to properties, by using concrete or virtual models to define, measure, observe, and change properties using models and/or technology to focus on defining properties, making property lists, and discussing sufficient conditions to define a shape doing problem solving, including tasks in which properties of shapes are important components classifying using properties of shapes.
3. Informal Deduction Students are able to recognize relationships between and among properties of shapes or classes of shapes and are able to follow logical arguments using such properties. Suggestions for instruction using informal deduction • doing problem solving, including tasks in which properties of shapes are important components • using models and property lists, and discussing which group of properties constitute a necessary and sufficient condition for a specific shape • using informal, deductive language ("all," "some," "none," "if-then," "what if," etc.) • investigating certain relationships among polygons to establish if the converse is also valid (e.g., "If a quadrilateral is a rectangle, it must have four right angles; if a quadrilateral has four right angles, must it also be a rectangle?") • using models and drawings (including dynamic geometry software) as tools to look for generalizations and counter-examples • making and testing hypotheses • using properties to define a shape or determine if a particular shape is included in a given set.
4. Deduction Students can go beyond just identifying characteristics of shapes and are able to construct proofs using postulates or axioms and definitions. A typical high school geometry course should be taught at this level. Note: Students usually do not reach Levels 4 and 5 until high school or college, butteachers should be aware of these levels nonetheless. Suggestions for instruction using deduction • using the components of an axiomatic system (undefined terms, definitions, postulates, theorems). • creating, comparing, and contrasting different proofs. • does NOT compare axiomatic systems.
5. Rigor This is the highest level of thought in the van Hiele hierarchy. Students at this level can work in different geometric or axiomatic systems and would most likely be enrolled in a college level course in geometry. Note: Students usually do not reach Levels 4 and 5 until high school or college, butteachers should be aware of these levels nonetheless. Suggestions for instruction using rigor • comparing axiomatic systems (e.g., Euclidean and non-Euclidean geometries). • rigorously establishing theorems in different axiomatic systems in the absence of reference models.
Summing up:What are the implications of van Hiele for instruction?
Van Hiele Match Activity Under the headings, match the correct boxes in the correct order.
Geometry Anticipation Guide a) In the first blank, mark “+” if you agree with the statement or “o” if you disagree with the statement. b) Next, when directed to do so, discuss the statements in your group. c) In the second blank, the class should discuss and decide on a “+” or a “o” for each blank. 6 – 8 will read the Overview, the critical areas, and the standards for grades 6 - 8. The high school will read the Conceptual Category: Geometry, the Geometry Overview, and all the Geometry standards. After reading, in the second blank, write whether you agree or disagree. If you disagree, change the statement so that you will agree with it and cite where the information came from in the document
Anticipation Guide for Geometry and van Hiele’s levels of Geometric Reasoning Using your notes on van Hiele’s model, discuss the level of geometric reasoning for each statement or statement that you changed on the anticipation guide. What can we do if students are not on that level? They may be above or below.
6 – 8 Geometry Domain and High School Geometry Conceptual Category Study High School Directions • The domains and clusters for the conceptual category Geometry are given. • Provide a description of how the domain is unique from the others. • Provide each domain and cluster that connects to the domain category that is being investigated. (The Geometry Overview will be helpful.) • Explain how each domain relates to the van Hiele model of geometric reasoning. 6th – 8th Grade Directions For grades 6, 7, and 8, the standards for each cluster are given. Provide a description of how the cluster is unique from the others. Provide each domain and cluster that connects to the domain category that is being investigated. (The overview for each grade will be helpful.) Explain how each cluster relates to the van Hiele model of geometric reasoning.
6 – 8 Geometry Domain and High School Geometry Conceptual Category Study For 6 – 8, what are the big ideas in the Geometry Domain for the 6th, 7th, and 8th grades? For high school, what are the big ideas in the Geometry Conceptual Category? After listing the big ideas on chart paper, discuss: - What is unique about each big idea? - What domains or clusters, in this grade or previous grades, support these big ideas? - What Van Hiele level of geometric reasoning will students need for these big ideas?
Modeling in Geometry:Modeling links classroom mathematics and statistics to everyday life, work, and decision making. Modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions. 2013 Revised Alabama Course of Study: Mathematics Beginning on page 76, highlight all ideas that relate to modeling in Geometry. Continue highlighting standards that are in the high school math courses, pages 88 – 120, that you believe relate to modeling. Provide a justification or example of why these standards are modeling standards. What standards in the 6th – 8th grades lead to these modeling standards? How are the van Hiele levels exemplified in these standards?
Reflection How did exploring the College and Career Ready standards for Geometry and van Hiele’s model deepen your understanding of the flow of the CCRS math standards? How might understanding a mathematical progression impact instruction? Give specific examples with respect to: • planning lessons • helping students make mathematical connections, • working with struggling students, • working with advanced students, and • using formative assessment and revising instruction
How can progressions in the standards within and across grades be used to inform teaching and learning? Would you teach the CCR standards in the Alabama COS, Mathematics in order? Standard #1, then Standard #2, then Standard #3, …
For teaching and learning in your classroom, how do you see connections between the content standards? What standard would you teach before or after another standard? When would you teach a particular standard during the year? At the beginning of the year In the middle of the year At the end of the year A B C Are some standards foundational for multiple other standards?
The Wire Diagram • Locate a standard in your grade band that requires students to build upon previous knowledge. High school teachers locate a standard in the 8th grade. • Determine what standards from earlier grades must be mastered in order for students to be successful in this new learning.
Why is professional peer discussion about progressions important for the teaching and learning process? “Analysis and related discussion with your team is critical to develop mutual understanding of and support for consistent curricular priorities, pacing, lesson design, and the development of grade-level common assessments.” Together you can develop a greater understanding of the intent of each content standard cluster and how the standards are connected within and across grades. (Common Core Mathematics in a PLC at Work, Kanold, 2012, pg. 67)
Geometry Through the GradesThis is a sample learning progression for 2D and 3D geometry. • 19 [K-G3] • 20 [K-G4] • 20 [1-G2] • 24 [2-G1] • 24 [3-G1] • 26 [4-G1] • 26 [5-G4] • 22 [6-G2] • 13 [7-G3] • 24 [8-G9] • Geometry 39 [G-MG1]; Precalculus 38 [G-GMD2]
Geometry Through the Grades • Using the list of standards that are related • to 2-D and 3-D geometry, discuss the following: • 1. What relevant changes occur from grade to grade? • Consider both content and practices. • How should the teacher accommodate students with different entry points? • Students that have not mastered the previous grade’s standard(s). • Students that have mastered the previous grade’s standard(s).
Toward Greater Coherence As you explore your assigned task, discuss the following; • Identify the Standards for Mathematical Practice that are central to the task. Are they handled in a grade-appropriate way, and are they well connected to the content being addressed? • How can teachers use the task or adjust it to provide appropriate level and type of scaffolding, differentiation, intervention and support for a broad range of learners? Does the task, or can it be modified to: • Support diverse cultural and linguistic backgrounds, interests and styles? • Provide extra supports for students working below grade level? • Provide extensions for students with high interest or working above grade level? • Should the teacher use this task as an assessment? Why or why not? What kind of assessment? • Where does the reasoning for this task align to on the van Hiele model?
Step Back – Reflection Questions • What are the benefits of considering coherence when designing learning experiences (lesson planning) for students? • How can understanding learning progressions support increased focus of grade level instruction? • How do the learning progressions allow teachers to support students with unfinished learning (struggling students) or advanced learning?
…. The Teacher Leader (AQTS 5.3) • How can today’s learning of the progressions be used to inform your teaching and learning? • How can today’s learning of the progressions be used to inform your professional learning community?
Wrapping up….. Prepare for District Team Planning
References • “The Structure is the Standards” Daro, McCallum, Zimba (2012) http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/ • www.illustrativemathematics.org • K–8 Publishers’ Criteria for the Common Core State Standards for Mathematics (2013)
