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Keep In Mind Map (Big Ideas (all relating back to) (Conceptual…

Keep In Mind Map

Planning

Chapter 7: From Tracking to Growth Mindset Grouping (Boaler)

Providing opportunities to learn for all students; ensuring high-level content is available for all students

De-Tracking (not forming separate classes providing higher or lower level content)

Providing optional advanced classes and extra math classes for everyone (not communicating fixed mindset messages)

Designing low-floor, high-ceiling tasks (broad range for a variety of students)

Chapter 5: Rich Mathematical Tasks (Boaler)

Incorporate curiosity, connection making, challenge, creativity, and collaboration into the classroom – increase engagement/excitement

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Allow for intuition 
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Design different, engaging tasks based on the makeup of the class (knowing your students and what works for them)

Connect related topics – bring ideas together

Planning Tasks (to make tasks richer):

1. "Can you open the task to encourage multiple methods, pathways, and representations?"

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2. "Can you make it an inquiry task?"
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3. "Can you ask the problem before teaching the method?"

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4. "Can you add a visual component?"
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5. "Can you make it low floor and high ceiling?"

6. "Can you add the requirement to convince and reason?"

Arbitrary and Necessary (Hewitt)

Distinguishing between arbitrary and necessary material prior to delivering a lesson

Givens can become known to a student in 3 ways (know which way you are going to present givens in a problem) - Student can receive, observe, or create their own givens

Limit received wisdom given, or, plan for more time for students to understand the received wisdom they’ve been offered (there is too much replication and moving on from material without fully understanding in the present school environment)

Chapter 8: Assessment for a Growth Mindset (Boaler)

Plan for open-ended projects – more conceptual understanding

Plan for “open” classrooms/safe environment

Planning how to bridge the gap between what students know and what they need to know

Chapter 6: Mathematics and the Path to Equity (Boaler)

Math is thought of as performance based (those who can and cannot) – need to look at it as available to all through study and hard work - focus on making it available to all

Offer all students high-level content (don’t offer low-level courses)

Work to change ideas about who can achieve in mathematics - Remove teacher judgment about incoming students; have a belief that all students can succeed (regardless of the past)

Planning activities that involve failure and struggle (knowing that it will help stimulate brain growth)

Chapter 9: Teaching Mathematics for a Growth Mindset (Boaler)

Planning group work with desirable group work behaviour

Think about the “didactic contract” - Refrain from playing into the expected role of helping students, without first giving the students an opportunity to learn

Instead of focusing on performing calculations, students need to also be setting up models, refining them, and using them to solve problems

Implementing

Chapter 7: From Tracking to Growth Mindset Grouping (Boaler)

Teaching Heterogeneous Groups Effectively

Provide open-ended tasks - low floor, high ceiling

Offer a choice of tasks (extra challenge)

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Individualized pathways
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Complex Instruction – to make group work equal:

1. "Provide multidimensional tasks - Students support each others learning"

2. "Assign roles" (everyone is important)

3. "Assign competence (praise an idea of a lower status" student)

4. "Teach students to be responsible for each other’s learning"

Explaining to peers – enhance each other’s understanding

Value what other students bring to the process (different strengths)

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Provide justification and reasoning 
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Promote relational equity
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Chapter 5: Rich Mathematical Tasks (Boaler)

Openness of numbers - Promote/encourage different ways of seeing and solving problems

Use Visualization Techniques and Tactile Manipulatives

Explain or tell methods after students explore a problem - Students learn methods after exploring the problem; methods are then taught when they are needed after deep thinking occurs

Arbitrary and Necessary (Hewitt)

Presenting arbitrary material to students (names, labels, symbols)

Helping students to work with adopted conventions

Allow students to work out the necessary from their prior knowledge

Provide a task for students to educate their own awareness (necessary); educate students’ awareness, rather than giving them something to memorize (arbitrary)

Allowing for creativity; maybe students will come up with arbitrary conventions (ex/ in Ch. 9 of Boaler: “Sean’s numbers”) – encourage this type of creativity

"For arbitrary, role is to assist memory"

"For necessary, role is to work within realm of awareness"

Limit the received wisdom given to students, or let students work out for themselves if received wisdom is true (they need to understand why it’s true)

Chapter 8: Assessment for a Growth Mindset (Boaler)

Shift teaching from worksheet math to inquiry-based math

Eliminate tracks so all students can be high achievers

Chapter 6: Mathematics and the Path to Equity (Boaler)

Refrain from referring to students as “gifted” – it promotes a fixed mindset where students are less likely to take risks

Change beliefs about math, student potential, and mindsets – reject the elitism of math

Shift the mindset away from “people who calculate quickly are smart and special” and towards valuing deep thinking

Move from performance to learning; embrace math as multidimensional

Requiring students to explain why – they will learn to understand and apply content

Transmit growth mindset beliefs to students

Encourage students to think deeply about mathematics - Teach math as a connected, inquiry-based subject

Give girls and students of colour additional encouragement to learn math and science:

1. "Replace sympathetic messages with positive ones"

2. "Highlight achievements of stereotyped and underrepresented groups in math" - Some students may face further barriers; work to address them

Praising failure and struggle – making the classroom a safe environment - Students will less likely avoid difficult situations and instead embrace them if they are not afraid of failing

Chapter 9: Teaching Mathematics for a Growth Mindset (Boaler)

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Encourage all students
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Set up classroom norms: Believe in everyone, Embrace mistakes, failure, & struggle, Value deep thinking, Encourage questions

Give growth praise and help - Focus on particular strategies students use instead of labeling them or using concrete language (ex/ "wrong," "smart")

Value and honour students’ thinking – there is always some logic in it

Teach math as an open, growth, learning subject: Encourage different ways of seeing math, representing ideas, Encourage different pathways and methods

Allow students to be mathematicians - Have them make conjectures and propose ideas

Teach math as a subject of patterns and connections: Connect material and content to meaningful experiences, Draw connections between different methods and concepts, Highlight the connectedness of math

Allow for creativity and visuals - Use colour coding and different ways to represent ideas

Encourage intuition and freedom of thought

Value depth over speed: “What is important is to deeply understand things and their relations to each other,” Deeper thinking – keeps students engaged and prepares them for future high-level learning, Extend problems

Connect math using mathematical modeling - Apply math to real-world situations and data; give context when helpful (Real-world – more engaging and realistic/applicable)

Have students pose questions, reason, justify, and be skeptical: Describe their steps and thought process to others to see if there are errors in their logic (they are reasoning), Explain their work

Teach with technology and manipulatives - Motivating students to make connections, Engaging and shows multidimensionality of math

Assessment

Chapter 7: From Tracking to Growth Mindset Grouping (Boaler)

Group work (but need to ensure there is equal participation)

Teach students to be responsible for each other’s learning - Peer assessment (learning from each other)

Chapter 5: Rich Mathematical Tasks (Boaler)

Include an element in assessment for students to convince and reason – demonstrating a conceptual understanding: Explaining their thought process and understanding why

Offering multiple forms of assessment (not just a test)

Including a visual component in assessment (representing their ideas in a different way; supporting their understanding)

Arbitrary and Necessary (Hewitt)

“If I’m having to remember, then I’m not working on mathematics”

Distinguishing between assessment of arbitrary and necessary: What are you trying to assess?

Arbitrary – realm of memory (ex/ definitions)

Necessary – realm of awareness (ex/ properties, relationships)

All students need to be informed of the arbitrary; the necessary is dependent upon the awareness students have - Not all students will have awareness to complete the necessary; someone is able to though

Received wisdom - "Students can become aware of the necessary if they work at it; then received wisdom becomes a derived certainty known through awareness rather than memory" (if they are understanding it)

Chapter 8: Assessment for a Growth Mindset (Boaler)

Students who worked through more open-ended projects – more successful on test questions they attempted

Grades and test scores demotivate rather than motivate students (damaging fixed messages, lower achievement)

Diagnostic comments only – students achieved at significantly higher levels

Assess less - Replacing grading with diagnostic comments – provide students with more insight

"Well-crafted tasks/questions and clear feedback offer students a growth mindset pathway" - Helps them know they can learn at high levels and how they get there

Change methods of assessment from fixed to growth - More intellectual freedom

Negative impact of grading – An “A student” will avoid harder work for fear of failure

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Intrinsic vs extrinsic motivation
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Extrinsic – motivated by scores and grades

Shift to intrinsic (interest in ideas they are learning, achievements increase)

“I remember the first time that a grading rubric was attached to a piece of my writing…Suddenly all the joy was taken away. I was writing for a grade – I was no longer exploring for me. I want to get that back. Will I ever get that back?”

Assessment for learning (Teachers students to take responsibility for their own learning)

1. Where students are now (what they know)

2. Where students need to be (what they need to know)

3. Ways to close the gap between the two)

Developing Student Self-Awareness and Responsibility - Reflection/metacognition/take control of own learning/know what they’re working on

Self-Assessment- Think about what they’ve learned and what they need to work on (reflecting), Accurate at assessing their own understanding

Peer Assessment: Assess each other’s work with clear criteria, Students much more open to criticism or suggestions from another student; Ex/ Two Stars and A Wish (important; did this during practicum)

Reflection Time- Reflect (and document) on big ideas, what they learned, questions they may have, how they can apply what they learned

Traffic Lighting - Prompts reflection for students and gives important information to teachers, Could use technology so students remain anonymous to other students

Jigsaw Groups - Working in groups, become experts on a subject, then teach others the knowledge they’ve learned, Taking responsibility of other’s learning – explaining to them

Exit Ticket - Giving ideas/information for next lesson (helps with planning process for next lesson)

Online Forms - Feedback to teacher during the lesson, Can get more feedback/assessment from students who do not usually participate verbally

Doodling - Connecting visual and intuitive mathematical thinking; drawing

Students write questions and tests - Enhances creativity; focus on what’s important

Diagnostic Comments - Help close gap between where students are and where they need to be, Feedback instead of scores

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Grading: 
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"Always allow students to resubmit any work or test for a higher grade" - Growth mindset; care about learning

"Share grades with administrators but not students" - Give students verbal or diagnostic written feedback

"Use multidimensional grading:" Use students math work rather than test performance

"Do not use a 100 point scale" (unrealistic)

"Do not include early assignments from math class in end-of-class grade" - Grades should show what students have learned

"Do not include homework as part of grading" - Inequitable practice

Change from grading and testing to formative assessment – powerful impact on students’ achievement, self-beliefs, motivation, and future learning pathways

Chapter 6: Mathematics and the Path to Equity (Boaler)

Teach students to work together: Work collaboratively and provide opportunities to see and understand connections, Learning from each other

Eliminate or change the nature of homework: Homework can increase inequity depending on the nature of the home, It does not adequately assess students’ understanding or knowledge