1. ASSESSMENT
1.1. Things to think about
1.1.1. By observing mathematical proficiency, teachers acknowledge student's ability, address difficulties, assist knowledge development and therefore can plan and structure their classes and activities focused on students' needs
1.1.2. Assessments should assess what's important
1.1.3. Math tasks and assessments should be interwoven
1.2. Assessment tools should:
1.2.1. Address POS in a balanced way
1.2.2. Encompass tasks that are acknowledged as worthwhile
1.2.3. Fit its purpose
1.2.4. Be challenged but still accessible
1.2.5. Focus on reasoning instead of results
1.2.6. Present tasks about genuine contexts
1.2.7. Encourage decision making
1.2.8. Be clear in terms of its demands
1.3. Assessment for learning
1.3.1. Parts of AFL:
1.3.1.1. Clearly communicating to students what they have learned
1.3.1.2. Helping students become aware of where they are in their learning journey and where they need to reach
1.3.1.3. Giving students information on ways to close the gap between where they are no and where they need to be
1.3.2. Formative assessment: informs learning and is the essence of assessment for learning
1.3.2.1. Chance for students to learn where they are at, what they need to know and how to close the gap between the two
1.3.3. Summative assessment: summarizes a student's learning
1.3.3.1. Gives final account of how far a student has gotten
1.3.4. Students are responsible for their own learning
1.3.4.1. Autonomous learners who can self-regulate and determine what and how to learn
1.4. When Grading...
1.4.1. Always allow students to resubmit any work or test for a higher grade
1.4.1.1. Ultimate growth mindset message
1.4.1.2. Care about learning, not performance
1.4.2. Share grades with school admin, but not with students
1.4.2.1. Give diagnostic comments instead
1.4.3. Use multidimensional grading
1.4.4. Be fair
1.4.4.1. Don't include early assignments from math class in the end-of-class grade
1.4.4.1.1. Should be grading what they learned in class, not what the learning from last year
1.4.4.2. Don't use a 100 point Scale
1.4.4.2.1. Unfair
1.4.4.3. Don't include homework
1.4.4.3.1. Homework perpetuates inequalities in education
1.4.4.3.2. Homework doesn't raise achievement in students
1.5. Grades
1.5.1. Ego feedback
1.5.1.1. Compare self to others
1.5.1.2. Damage to learning
1.5.1.3. Students define themselves by their grades
1.5.1.4. Demotivates students
1.5.2. Communicate fixed and damaging messages
1.5.2.1. Lower classroom achievement
1.5.2.2. Fixed mindset
1.5.3. Extrinsic motivation: motivation from the thought of getting better grades and scores
1.6. Grade + Comments
1.6.1. Students only focus on grades
1.7. Comments
1.7.1. Growth mindset
1.7.1.1. Talk to student about their mistakes
1.7.1.1.1. Let them learn from them
1.7.1.2. Students are always growing and learning
1.7.2. Empower learners
1.7.3. Intrinsic motivation: comes from interest in the subject and ideas
1.7.4. Diagnostic Comments
1.7.4.1. Help students know what they are learning and where they should be in their learning
1.7.4.2. Help student understand how to close the gap of where they are and where they need to be
1.7.4.3. Growth messages
2. PRE-PLANNING
2.1. Things to Think About
2.1.1. Wrong Assumptions
2.1.1.1. Separate students with from students without the math gene
2.1.1.2. Culture of performance and elitism
2.1.2. Good Considerations
2.1.2.1. Keep expectations and opportunities high for all students
2.1.3. Equitable Strategies
2.1.3.1. Offer all students high-level content
2.1.3.2. Change ideas about who can achieve in mathematics
2.1.3.3. Encourage students to think deeply about mathematics
2.2. Math
2.2.1. An incredible lens
2.2.2. For important knowledge
2.2.3. Available through study and hard work
2.2.4. Promotes empowered young people who are ready to think quantitatively about their work and lives
2.2.5. Connected, inquiry based subject
2.2.6. Mathematical proficiency
2.2.6.1. Conceptual understanding
2.2.6.1.1. Speaks to students’ retrieval, connection and comprehension of mathematical ideas, content and representations. Allows students to better retain and (re)construct knowledge, gets students ready to detect conceptual errors, requires students to “learn less” (once content is interrelated), and provides students with confidence.
2.2.6.2. Strategic competence
2.2.6.2.1. Involves formulating, representing and solving problems. Allows students to realize similarities in tasks’ structure, to identify relations between mathematical elements, and to figure out different solving models or representations when faced with non-routine situations, which stimulates flexibility and productive thinking.
2.2.6.3. Procedural fluency
2.2.6.3.1. Refers to the appropriate and flexible use of procedures, as well as to the comprehension of them. Without enough procedural fluency, procedures may be compartmentalized, students’ mathematical understanding may be superficial, and problem solving skills may be impaired.
2.2.6.4. Adaptive reasoning
2.2.6.4.1. Alludes to the ability of building logical connections between mathematical ideas, contents and circumstances. It is not only about formal proofs, but also about argumentation, justification and reasoning.
2.2.6.5. Productive disposition
2.2.6.5.1. It is necessary if students are to develop the other four strands; and it is determinant on students’ academic achievement. It is about acknowledging and believing in the benefits of mathematics, and about trusting in one’s own ability to do and learn mathematics.
2.3. Myth of the mathematically gift child
2.3.1. Labeling students hurts students
2.3.1.1. Course placement = a source of inequalities
2.3.2. Fixed mindset
2.3.2.1. More vulnerable
2.3.2.2. Less likely to take risks
2.4. Drawing Upon and Incorporating Indigenous Knowledge in The Classroom
2.4.1. Key areas for transformation:
2.4.1.1. The need to learn the language
2.4.1.1.1. Dominant starting point
2.4.1.1.2. Everything is linked to language
2.4.1.1.3. Awareness of mathematical concepts that have no translation in the indigenous language
2.4.1.2. The importance of attending to value differences between indigenous concepts of math and school based math
2.4.1.2.1. Ideas of 'enough', number as play and the importance of space
2.4.1.3. The importance of attending to ways to learning and knowing, approaches rooted in spatial reasoning
2.4.1.4. The significance of making ethnomathematical connections for students
2.4.2. Verbifying mathematics is an approach that draws upon the grammatical structures in indigenous language
2.4.3. Cultural connections
2.4.3.1. Essential to make meaningful and non-trivializing connections to the community and culture
2.4.3.2. Create learning experiences that help students see mathematical reasoning in a part of their everyday life
2.4.3.3. Let indigenous knowledge take a central role in math class without simply imposing western math on cultural artefacts
3. PLANNING
3.1. Opportunity to learn = key
3.1.1. Low floor, high ceiling tasks
3.2. Have a clear goal
3.2.1. Focus on essential understandings
3.2.2. Building conceptual connections with prior knowledge, emphasizing meaning, requiring explanations, thinking processes and strategies are essential for maintaining engagement
3.3. Unit Planning Components:
3.3.1. Learning Goals
3.3.1.1. Identifies knowledge and skills for the POS
3.3.1.1.1. Focuses around the specific skills students would need to meet the unit goals
3.3.1.2. Incremental and scaffolded
3.3.1.3. Expressed in language meaningful to students
3.3.1.3.1. Assessment as learning = student monitoring their own learning
3.3.1.4. Uses clear, concise language
3.3.1.5. Specific and observable
3.3.1.6. Stated from the student's perspective
3.3.1.7. Based on POS
3.3.2. Anticipated struggles and prior knowledge
3.3.2.1. Helps plan resources and supports
3.3.3. Planned supports
3.3.3.1. Relevant resources
3.3.3.2. Connected to anticipated struggles and prior knowledge
3.4. Lesson Planning
3.4.1. Adaptable, critical, and analytical thinking is needed by students for the modern world
3.4.2. Establish a clear purpose and objectives for both teachers and students
3.4.3. Lessons should connect with prior knowledge, capture interest, and provide chances for meaningful practice in and out of the class
3.4.4. Task design:
3.4.4.1. Address genuine, engaging and challenging situations
3.4.4.1.1. Prompt the development of different abilities
3.4.4.2. Demand mathematical curriculum knowledge
3.4.4.3. Require student's autonomy and decision making
3.4.4.4. Focus on reasoning
3.4.4.5. Acknowledge mathematics worthiness
4. TEACHING
4.1. Teaching Effectively
4.1.1. Providing open ended questions
4.1.1.1. Engaging and interesting
4.1.1.2. Inspire interest
4.1.1.3. Encourage creativity
4.1.2. Offer a choice of tasks
4.1.3. Individualized pathways
4.2. Developing student's self awareness and responsibility
4.2.1. Self assessment:
4.2.1.1. Become aware of the math they are learning and their broader learning pathways
4.2.2. Peer assessment:
4.2.2.1. Give clear criteria to assess the work
4.2.2.2. Gain additional chances to become aware of the math you are learning and what you need to learn
4.2.3. Reflection time
4.2.3.1. Provide the time at the end of a lesson
4.2.3.1.1. E.g. Exit ticket
4.2.3.2. Traffic lighting
4.2.3.2.1. Indicate where the students is in terms of their learning
4.2.3.3. Online forms
4.2.4. Jigsaw groups
4.2.4.1. Students working together to become experts on a particular topic, sections, etc.
4.2.4.2. Students are responsible for teaching others --> take responsibility for the new knowledge they are learning
4.2.5. Student write questions and tests
4.2.5.1. Help students focus on what's important
4.2.5.2. Interesting and engaging
4.2.5.3. Creative
4.3. Student engagement is based on different factors
4.3.1. The work of the teacher
4.3.1.1. Setting up tasks
4.3.1.2. Helping students when they're stuck
4.3.2. The task given
4.3.2.1. Open and challenging
4.3.3. The multidimensionality of the classroom
4.3.3.1. Different ways to do math
4.3.4. Request to deal with a real-world object and idea
4.3.5. High levels of communication
4.3.5.1. Students and the teacher support each other
4.3.5.2. Self and group responsibility for learning
4.4. Learning in a second language is possible
4.4.1. Need the right support and resources
4.4.1.1. Takes several years to develop academic language skills in a second language
4.4.1.1.1. Important to academic performance
4.4.2. Math is a complex language
4.4.2.1. Requires the understand of...
4.4.2.1.1. Logical connectives
4.4.2.1.2. The coordination of syntax and logic
4.4.2.1.3. Definitions, conjectures, questions, and explanations
4.4.3. Language Production (speaking and writing) Just as important as hearing or reading for the learning process
4.4.3.1. Prompts deeper, more focused processing of the target language
4.4.3.2. Meaningful dialogue = effective exposure
4.4.3.2.1. Multiple ways to talk about a mathematical idea
4.4.3.2.2. Rewording student's ideas using more conventional mathematical language
4.4.3.2.3. Make sure the focus of the discussion remains on the mathematical ideas
4.5. Mathematical meaning making from:
4.5.1. Personal experiences
4.5.2. Their culture, families, friends and other languages they know
4.5.3. Their knowledge of math and mathematical language
4.5.4. Deixis- this one, that one
4.5.5. Multiple modes of representation
4.5.5.1. Concrete materials, graphs, diagrams, models, symbols, gestures, writing, etc.
4.5.6. Different types of talk
4.5.6.1. E.g. Expository, exploratory
4.5.6.2. Mathematical Discussions
4.5.6.2.1. Listen to make sense
4.5.6.2.2. Pose questions to provoke further thinking
4.5.6.2.3. Reword using more mathematical language (written or spoken)
4.5.6.2.4. Drawing attention to important math ideas
4.5.6.2.5. Drawing attention to important math language