Mathematical Mindsets

1. The brain's plasticity enables growth and change at any age

When we learn a new idea, one of three things happens in the brain... The first possibility is that you start a new brain pathway. The more deeply you learn, the stronger the pathway becomes.

Brain plasticity revolutionizes learning. Neuroscience has shown that our brains are incredibly adaptable, capable of forming new connections and pathways throughout our lives. This challenges the traditional notion that mathematical ability is fixed or innate.

Key findings on brain plasticity:

• London taxi drivers' hippocampi grow with spatial learning
• Brain changes occur even with brief, 10-minute daily exercises
• Students with math learning disabilities show brain changes after tutoring

The implications for mathematics education are profound. It means that all students, regardless of age or background, have the potential to develop strong mathematical skills. Teachers and parents should encourage a growth mindset, emphasizing that mathematical ability can be developed through effort and practice.

2. Mistakes and struggle are essential for mathematical learning

Every time a student makes a mistake in math, a synapse fires.

Mistakes fuel brain growth. Research has shown that when we make mistakes, our brains experience increased electrical activity. This activity occurs even if we're unaware of the error, indicating that the process of struggling with a problem is itself valuable for learning.

Benefits of mistakes in mathematics:

• Increased brain activity and synapse firing
• Development of problem-solving skills
• Opportunity for deeper understanding

Educators should create environments where mistakes are welcomed and celebrated. This shift in perspective can help reduce math anxiety and encourage students to take intellectual risks. By framing struggle as a positive and necessary part of learning, we can help students develop resilience and a love for mathematical challenges.

3. Mathematics is a creative, visual, and connected subject

Mathematics is a cultural phenomenon; a set of ideas, connections, and relationships that we can use to make sense of the world.

Math is everywhere. Contrary to the common perception of mathematics as a rigid, procedural subject, it is fundamentally about patterns, creativity, and connections. Mathematics exists throughout nature, art, and the world around us.

Examples of mathematics in nature and culture:

• Fibonacci sequence in flower petals and pinecones
• Fractals in snowflakes and coastlines
• Mathematical models in computer-generated special effects
• Data analysis in criminal investigations

By presenting mathematics as a broad, multidimensional subject, educators can engage more students and help them see its relevance to their lives. Encouraging visual thinking, pattern recognition, and creative problem-solving can make mathematics more accessible and enjoyable for all learners.

4. Developing a growth mindset is crucial for mathematical success

Students with a growth mindset keep going even when work is hard and are persistent, displaying what Angela Duckworth has termed "grit".

Mindset shapes achievement. Research by Carol Dweck and others has shown that students' beliefs about their own abilities significantly impact their mathematical performance. Those with a growth mindset – the belief that intelligence and ability can be developed through effort – consistently outperform those with a fixed mindset.

Key aspects of a mathematical growth mindset:

• Belief that everyone can learn mathematics to high levels
• Valuing effort and perseverance over innate "talent"
• Seeing mistakes as opportunities for learning
• Embracing challenges as chances for growth

Educators and parents can foster growth mindsets by praising effort and strategies rather than intelligence, encouraging risk-taking, and framing struggles as normal and productive parts of the learning process.

5. Traditional teaching methods often hinder mathematical understanding

Mathematics has been pulled into a culture of performance and elitism in the United States, and I believe that to achieve higher and equitable outcomes we need to recognize the elitist role that mathematics often plays in our society.

Rethink math education. Many traditional approaches to teaching mathematics, such as heavy emphasis on memorization, timed tests, and rigid procedural methods, can actually impede deep understanding and enjoyment of the subject.

Problems with traditional math education:

• Overemphasis on speed and memorization
• Narrow focus on procedural fluency at the expense of conceptual understanding
• Use of tracking and ability grouping, which can reinforce fixed mindsets
• Reliance on pseudo-contexts that don't reflect real-world math applications

Instead, educators should focus on developing conceptual understanding, encouraging multiple solution strategies, and providing opportunities for creative problem-solving. This approach aligns more closely with how mathematicians actually work and helps students develop the flexible thinking skills needed in the modern world.

6. Equitable mathematics education requires dismantling harmful stereotypes

Mathematics is the subject most in need of a mindset makeover.

Math is for everyone. Persistent stereotypes about who can excel in mathematics – often based on gender, race, or socioeconomic background – create significant barriers to equitable education. These biases can lead to self-fulfilling prophecies, where students internalize negative beliefs about their mathematical abilities.

Strategies for promoting equity in mathematics:

• Challenging stereotypes about mathematical ability
• Providing role models from diverse backgrounds
• Using inclusive language and examples in teaching
• Offering high-level content to all students
• Implementing unbiased assessment practices

Educators must actively work to counteract these harmful stereotypes by providing encouragement and high expectations for all students, regardless of background. By creating inclusive classroom environments and using diverse examples and contexts, we can help all students see themselves as capable mathematicians.

7. Effective assessment focuses on learning, not performance

When students are given a percentage or grade, they can do little else besides compare it to others around them, with half or more deciding that they are not as good as others.

Rethink assessment practices. Traditional grading and testing methods often promote a fixed mindset and can be detrimental to student learning. Instead, assessment should be used as a tool for growth and improvement.

Principles of effective mathematical assessment:

• Providing specific, actionable feedback rather than just grades
• Using formative assessment to guide instruction
• Encouraging self-assessment and reflection
• Allowing multiple attempts and opportunities for improvement
• Focusing on understanding and problem-solving, not just correct answers

By shifting the focus from performance to learning, teachers can help students develop a growth mindset and take ownership of their mathematical development. This approach also provides more meaningful information to guide instruction and support student progress.

8. Rich mathematical tasks promote engagement and deeper understanding

When mathematics tasks are opened for different ways of seeing, different methods and pathways, and different representations, everything changes.

Design engaging math tasks. The quality of mathematical tasks has a significant impact on student learning and engagement. Rich, open-ended tasks that allow for multiple approaches and representations can deepen understanding and promote mathematical thinking.

Characteristics of effective mathematical tasks:

• Low floor, high ceiling (accessible to all, with room for extension)
• Multiple entry points and solution strategies
• Opportunities for visualization and representation
• Connections to real-world contexts or mathematical ideas
• Encouragement of reasoning and justification

By using tasks that embody these principles, teachers can create more engaging and intellectually stimulating math classrooms. These tasks allow students to experience mathematics as a creative, problem-solving discipline rather than a set of procedures to memorize.

9. Collaboration and communication are vital in mathematics learning

Mathematics is a very social subject, as proof comes about when mathematicians can convince other mathematicians of logical connections.

Math is a team sport. Contrary to the stereotype of the solitary mathematician, collaboration and communication are essential in mathematical practice. Group work and mathematical discussions can significantly enhance learning and understanding.

Benefits of collaborative mathematics:

• Exposure to diverse problem-solving strategies
• Development of communication and reasoning skills
• Increased engagement and motivation
• Opportunity to learn from peers' mistakes and insights

Educators should create opportunities for students to work together, explain their thinking, and critique each other's reasoning. This not only deepens mathematical understanding but also prepares students for the collaborative nature of many STEM careers.

10. Teachers play a critical role in fostering mathematical mindsets

Teachers are the most important resource for students. They are the ones who can create exciting mathematics environments, give students the positive messages they need, and take any math task and make it one that piques students' curiosity and interest.

Teacher mindset matters. The beliefs and attitudes of teachers have a profound impact on student learning and mindset development. Teachers who believe in their students' potential and create supportive, engaging learning environments can transform students' mathematical experiences.

Key teacher practices for fostering mathematical mindsets:

• Communicating high expectations for all students
• Emphasizing effort and process over innate ability
• Providing opportunities for productive struggle
• Modeling a growth mindset and enthusiasm for mathematics
• Using diverse and engaging teaching strategies

Professional development that helps teachers develop their own mathematical mindsets and effective teaching practices is crucial for improving mathematics education. By empowering teachers to create positive, growth-oriented math classrooms, we can help all students develop strong mathematical identities and skills.

Last updated: January 24, 2025