Setting Up Classroom Norms:
- Positive Norms for Math Classes:
- Everyone can learn math to the highest levels.
- Mistakes are valuable.
- Questions are really important.
- Math is about creativity and sense making.
- Math is about connections and communicating.
- Depth is much more important than speed.
- Math class is about learning, not performing.
- In small groups, lets students specify norms for things they’d like to see / hear / experience (or not) while team problem solving. Create posters of student preferences.
- Skills to teach:
- Listening to each other
- Respecting each other
- Building on each other’s ideas
- Communicate expectations for what math looks like when teams are actively processing such as:
- Your group will be successful today if you are …
- Recognizing and describing patterns
- Justifying thinking and using multiple representations
- Making connections between different approaches and representations
- Using words, arrows, numbers and color coding to communicate ideas clearly
- Explaining ideas clearly to team members and the teacher
- Asking questions to understand the thinking of other team members
- Asking questions that push the group to go deeper
- Organizing a presentation so that people outside the group can understand your group’s thinking
- I will be looking for:
- Learning and working in the middle of the table
- Equal air time
- Sticking together
- Listening to each other
- Asking each other lots of questions
- Following your team roles
- You can use the checklists above to record what you’re observing while students work in teams and to provide them feedback on their teamwork.
- Your group will be successful today if you are …
Believe in All of Your Students
- Have high expectations for all students and provide support and positive messaging that helps students believe & demonstrate that they can achieve your high expectations
- Avoid early tracking
- Avoid unspoken messages that communicate that you don’t believe is someone’s potential – like only assigning them easy work
Value Struggle and Failure
- Assign challenging math problems that provide opportunities for struggle and learning mistakes
- Assign low floor, high ceiling tasks
- Communicate frequently that struggle and failure are good (failing forward)
- Break the myth of “effortless achievement”; all achievers worked hard and failed, even geniuses
Give Growth Praise and Help
- Growth praise and help focuses on strategies and effort, not on ability
- When students do math problem wrong – start by validating the strategy they used to first tackle the problem before redirecting them to new strategies
- Instead of breaking down problems for students – ask them to draw the problem and see what ideas come out of that activity
- When students can handle a little more struggle – respond to their requests for help by saying – Do you want my brain to grow or your brain to grow?
- Show students that math is a growth subject
Teach Math as an Open, Growth, Learning Subject
- Closed math problems – just ask for calculations, promote a fixed mindset
- Open up math problems so they invite students to think and grow.
- Example of opening up math problems:
- Closed form: What is 1/2 divided by 1/4?
- Open form: Make a conjecture about the answer to 1/2 divided by 1/4 and make sense of the answer by using a visual representation of the solution.
- Closed form: Simplify (1/3)(2x+15)+8
- Open form: Find as many ways as possible to represent (1/3)(2x+15)+8 that are equivalent.
- Closed form: Find the 100th case.
- Open form: How is the pattern growing? Use your understanding of the pattern to generalize to the 100th case.
- Example of opening up math problems:
- Ask students to discuss:
- ways of seeing mathematics
- ways of representing ideas
- different pathways through problems and solutions
- why use different methods
- how do different methods work
- Instead of just finding answers allow students to:
- explore ideas
- make connections
- value growth and learning
- learn standard procedures when they are ready to see the need for them and can make sense of them
Encourage Students to be Mathematicians
- What mathematicians do and think:
- math is creative, beautiful and aesthetic
- propose and test ideas
- develop working definitions for ideas based on consensus and reasoning
- share thinking and ideas
- Do not be afraid to call students, young mathematicians – why not? if they can be young artists and young musicians, why not young mathematicians?
Teach Mathematics as a Subject of Patterns and Connections
- Encourage students to see themselves as pattern seekers
- Teach traditional procedures as one of many sense making approaches to perform operations
- Encourage students to see math as a classification and study of all possible patterns
- Give students an active role in pattern seeking
- Help and let students see the connections between methods
Teach Creative and Visual Mathematics
- In expectations ask students not for speed, but for creative solutions to problems
- Engage students by asking them to represent problems visually
- Connect visual ideas with numerical or algebraic methods / solutions
- Color code:
- represent the same ideas (ex: the variable x) using the same color
- illustrate division by using different colors for partitions (division quilt)
Encourage Intuition and Freedom of Thought
- Encourage intuition by asking students what they think would work before showing them a method
- give them opportunities to try their methods on problems before teaching new methods
- Start with the hypothesis that any subject can be taught effectively in some “intellectually honest” form to a child (Bruner)
Value Depth over Speed
- Ask questions that are open enough to bring depth into discussions
- Closed form: Supplementary angles add up to what number?
- Open forms: Can two acute angles be supplementary angles? Can two obtuse angles be supplementary angles?
- Closed form asks for a single answer.
- Open form provokes conjectures and discussions.
- Ask students who finish early to extend problems in any way they wish
- Aim for depth, not speed – engage students by allowing them to go deeper into problems
Connect Mathematics to the World Using Mathematical Modeling
- Textbooks oven cast math in pseudo contexts (fake real world problems)
- Use real world variables part of the time to expose students to real uses of math
- View math as a posing questions and form math models around those questions
- Modeling – simplification of any real world problem into a pure math form that can help solve a problem
- Students often use modeling all the time, but are unaware of it
- Use visual representations to represent problems (one type of modeling)
- Use real data from newspapers, magazines, online databases, etc.
- Make students think about how contexts constrain possible solutions
Encourage Students to Pose Questions, Reason, Justify and Be Skeptical
- Offer students opportunities to pose questions to situations
- Example: Give students priced for finished bracelets and for bracelet supplies. Then ask them discuss the situation and pose questions.
- Give students opportunities to try out their own conjectures and use reasoning and data to prove or disprove them.
Teach with Cool Technology and Manipulatives
- Manipulatives: Cuisenaire rods, multilink cubes, pattern blocks
- Apps: Geometry Pad (iPad), GeoGebra, Tap Tap Blocks, and many more. See Rich Mathematical Tasks for ideas.
Teaching mathematics in ways that promote growth mindsets enables students to perceive math as a living, engaging, relevant, and accessible subject. Giving them access to mathematical processes other than computation, gives students a better chance to experience mathematics more fully and to relate mathematics to processes they already do – such as make conjectures, ask questions, and notice and use patterns. Creating the math culture that promotes growth mindset involves teaching students how to collaboratively problem solve, modeling and teaching more math phases (question formulation, modeling, computation, evaluating models, etc.), designing and facilitating math problems with low floors, high ceilings and authentic contexts, and integrating real world data and technology into problem solving.
Preparation Steps
- Design and implement learning activities that promote growth mindset and collaboration norms. Create visuals to market the ideas that emerge from these activities.
- Research and design curriculum that includes Rich Mathematical Tasks with low floors, high ceilings, open questions, and engaging (if possible real) contexts.
- Develop assessment tools that relate to math learning and collaboration processes.
- Research technology tools and manipulatives that can be used to create and facilitate more rich mathematical tasks.
Early Implementation Steps
- Implement the curriculum and tools planned above.
- Use student feedback to adjust learning experiences as needed.
Advanced Implementation Steps
- Identify helpful strategies that can be incorporated into routines to consistently promote a culture of growth mindset.
- Collaborate with other math teachers and teachers in related disciplines on norms and strategies that can be used in multiple contexts to cultivate growth mindset.
- Rich Mathematical Tasks
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- Mathematics articles