Mathematical Mindsets – What They Are:
- Present naturally in children who like to inquire, build things, solve puzzles, notice/make patterns, etc
- Seeing math as a conceptual subject that they can grow to make sense out of
- Stepping forward deliberately and deeply in math while making sure each step makes intuitive sense before moving more forward
How to Stunt Development of Mathematical Mindsets:
- Presenting math as a dry set of methods can stop development of mathematical mindsets
- this is especially true if methods do not make intuitive sense to students
- Presenting math as seires of short questions obscures growth opportunities – math is something you get or you don’t, instead of something to make sense of
- Assigning large homework sets with simple isolated problems
- Valuing rote memorization and speed over deep thinking and conceptual understanding
How to Develop Mathematical Mindsets:
- Encourage students to play with numbers, shapes and puzzles
- Present math as a broad landscape of unexplored puzzles that create opportunities for wandering around, asking questions, thinking of relationships, …
- Present math as a flexible conceptual subject that is about thinking and sense making
- Be mindful when designing practice set because mindless practice does not lead to brain growth, thoughtful practice does – mindful practice involves applying same strategy to many different situations
- Assign less homework that requires more reflection – example 5 carefully selected problems and one student chosen reflection question such as:
- What are the main mathematical ideas we discussed in class today?
- What questions do you have about ________?
- Describe a mistake or misconception you or a student had in class today. What did you learn from this mistake or misconception?
- How did you approach your practice set? Was your approach successful? What did you learn from your approach?
- Cultivating Number Sense:
- Approach arithmetic operations flexibly and conceptually:
- concept of sum -> counting on
- concept of product -> repeated addition
- Try to help students make sense of concepts and patterns so that their brain can more readily go from compression – more efficient storage of concepts (not rules)
- Math facts are stored in working area of brain – this area can be blocked when students are stressed
- Avoid techniques that value speed of knowing math facts (example – timed tests)
- Do NOT emphasize rote knowledge and speed – gets in the way of thinking about numbers and their relationships to each other
- Teach strategies instead of memorization of facts
- example: 17 x 8
- strategy – 17 x 10 – 17 x 2 = 170 – 34 = 136
- memorize 17 x 8 = 136
- example: 17 x 8
- play math games that activate both sides of brain by using visual and intuitive math thinking:
- example: grid multiplication game
- object of game – fill as many grid squares as possible in a 10 x10 grid
- roll 2 number dice – color in area that corresponds to product of 2 numbers rolled and write number sentence
- partners take turns rolling dice, coloring in areas and writing related number sentences until no more arrays can be added to the grid
- example: multi rep matching game
- players take turns picking pairs of equivalent cards and explaining why they are equivalent
- find cards and more cool strategies here
- example: grid multiplication game
- Do “Number Talks” as warmups
- pose a tough problem that can be solved multiple ways
- ask students to solve it using mental math
- have students present their mental math solutions
- Books / Videos with Number Talks:
- Recommend math games that emphasize concepts over drill & kill:
- Approach arithmetic operations flexibly and conceptually:
Developing mathematical mindsets will help student approach mathematics with a growth mindset. Mathematical mindsets help students understand math concepts more deeply and apply them more flexibly. Valuing conceptual understanding over speedy rote memorization is one way to cultivate mathematical mindsets.
Preparation Steps
- Research more strategies for developing mathematical mindsets. See Mathematics articles for ideas.
- Develop lesson plan components (Warmups, practice sets, discussions, activities, etc) that promote mathematical mindsets
Early Implementation Steps
- Regularly use scaffolding and assessments that promote mathematical mindsets
- Have students reflect often on what they are learning
- about concepts
- about how concepts are applied to problem solving
- from mistakes
- from different problem solving approaches
Advanced Implementation Steps
- Have students interact with mathematicians and professionals who apply mathematical reasoning often and learn about their problem solving approaches
- Develop bank of Number Talk problems and games that promote mathematical mindsets and incorporate these into classroom routines
- Mathematics articles
- Understanding articles