What's Math Got to Do with It?

1. Mathematics is about patterns, connections, and problem-solving, not just procedures

Mathematics is a human activity, a social phenomenon, a set of methods used to help illuminate the world, and it is part of our culture.

Mathematics is creative and interconnected. It involves exploring patterns, making connections between ideas, and solving complex problems. Real mathematical work is about inquiry, reasoning, and discovery - not just memorizing formulas and procedures. Mathematicians approach problems flexibly, trying multiple methods and representations.

Math is everywhere in the world around us. From the spirals in flowers to the algorithms in our technology, mathematics helps us understand and describe patterns in nature, science, art, music, and more. Learning to see and analyze these patterns develops powerful thinking skills applicable far beyond the math classroom.

Key aspects of authentic mathematical work include:

• Asking questions and posing problems
• Looking for and analyzing patterns
• Making conjectures and developing logical arguments
• Connecting ideas across different areas of math
• Applying mathematical thinking to real-world situations
• Communicating mathematical ideas clearly

2. Traditional teaching methods often fail to engage students and develop true understanding

In math you have to remember; in other subjects you can think about it.

Traditional methods emphasize memorization over meaning. Many math classrooms rely heavily on lecturing, rote practice of procedures, and timed tests. This approach often leaves students feeling that math is a set of arbitrary rules to memorize rather than a meaningful, interconnected subject. As a result, many students develop math anxiety and lose interest.

Procedural fluency without conceptual understanding is limiting. While students may be able to execute procedures correctly, they struggle to apply their knowledge to new situations or explain the reasoning behind methods. This surface-level learning doesn't develop the problem-solving and analytical skills that are truly valuable.

Problems with the traditional approach include:

• Overemphasis on speed and memorization
• Lack of opportunities for discussion and collaboration
• Limited connections to real-world applications
• Narrow focus on procedural knowledge
• Anxiety-inducing timed tests and high-stakes assessments
• Fixed ability mindsets that limit student potential

3. Effective math instruction emphasizes inquiry, discussion, and multiple representations

If you can tap into the real thoughts of the person before you, you can untangle the knots around their mathematical inner light.

Active learning develops deeper understanding. Effective math instruction engages students in exploring concepts through inquiry, discussion, and problem-solving. This approach helps students build connections between ideas and develop true mathematical thinking skills. Teachers guide students to construct their own understanding rather than passively receiving information.

Multiple representations illuminate concepts. Using varied representations - like diagrams, graphs, physical models, and verbal descriptions - helps students grasp mathematical ideas from different angles. This develops flexibility in thinking and problem-solving. Encouraging students to create their own representations deepens understanding.

Key elements of effective math instruction:

• Open-ended problems with multiple solution paths
• Collaborative group work and mathematical discussions
• Connections to real-world contexts and applications
• Visual models and manipulatives
• Emphasis on mathematical reasoning and justification
• Opportunities for students to pose their own questions

4. Ability grouping and tracking can limit student potential and reinforce inequalities

You're putting this psychological prison around them. . . . People don't know what they can do, or where the boundaries are, unless they're told at that kind of age.

Tracking often becomes a self-fulfilling prophecy. When students are placed in lower tracks, they typically receive less challenging work and lower expectations. This limits their opportunities to develop higher-level thinking skills. Students internalize these labels, affecting their confidence and motivation.

Mixed-ability grouping can benefit all students. Research shows that heterogeneous grouping, when implemented effectively, can raise achievement for students at all levels. Lower-achieving students benefit from exposure to more advanced ideas, while higher-achieving students deepen their understanding by explaining concepts to others.

Problems with ability grouping:

• Reinforces fixed mindsets about mathematical ability
• Often based on limited or biased assessment data
• Disproportionately affects minority and low-income students
• Limits access to higher-level coursework and college preparation
• Reduces opportunities for diverse perspectives in problem-solving
• Can lead to social segregation and reduced expectations

5. Timed tests and speed pressure negatively impact math learning and attitudes

Standardized testing has swelled and mutated, like a creature in one of those old horror movies, to the point that it now threatens to swallow our schools whole.

Timed tests induce anxiety and block working memory. Brain research shows that the stress of timed conditions interferes with students' ability to access mathematical knowledge. This leads to underperformance that doesn't accurately reflect true understanding. Timed tests also reinforce the misconception that speed equals mathematical ability.

Emphasis on speed narrows the curriculum. When teachers feel pressure to prepare students for timed standardized tests, they often focus on drilling procedures rather than developing deeper conceptual understanding and problem-solving skills. This limits students' mathematical growth and enjoyment of the subject.

Negative impacts of timed testing:

• Induces math anxiety, even in high-achieving students
• Disadvantages students who process information more slowly
• Reinforces procedural thinking over conceptual understanding
• Narrows the curriculum to focus on easily tested skills
• Discourages persistence on challenging problems
• Gives a distorted picture of mathematical proficiency

6. Girls and women face unique barriers in mathematics that must be addressed

Mathematics is one of the reasons girls do not go forward in STEM because girls look for a depth of understanding that is often unavailable in math classrooms.

Gender stereotypes impact confidence and participation. Societal messages that math is for boys can undermine girls' confidence and interest from an early age. Even well-intentioned statements like "girls are good at language" implicitly suggest they are not good at math. These stereotypes affect course choices, career aspirations, and persistence in STEM fields.

Teaching approaches often don't align with girls' learning preferences. Research shows that many girls prefer collaborative, discussion-based learning that emphasizes understanding over memorization. Traditional math instruction that focuses on speed and competition can be particularly alienating for girls.

Strategies to support girls in mathematics:

• Challenge gender stereotypes about mathematical ability
• Provide female role models in math and STEM careers
• Emphasize conceptual understanding and real-world applications
• Encourage collaborative problem-solving and discussion
• Address issues of belonging and identity in math classes
• Ensure equitable participation and recognition in class

7. Parents and teachers can foster mathematical thinking through puzzles, games, and open-ended exploration

Mathematical settings need not be sets of objects. They can be simple arrangements of patterns and numbers in the world around us.

Everyday activities can develop mathematical thinking. Parents don't need advanced math knowledge to support their children's mathematical development. Simple activities like counting, measuring while cooking, or looking for patterns in nature all build foundational skills. Games involving strategy, spatial reasoning, or probability naturally engage mathematical thinking.

Open-ended exploration sparks curiosity. Rather than always directing children's mathematical activities, adults can provide interesting materials or contexts and allow children to pose their own questions and explore. This develops problem-solving skills and helps children see math as a creative, inquiry-based subject.

Ways to encourage mathematical thinking at home:

• Play strategy games like chess, Set, or Blokus
• Explore patterns with building blocks or tangrams
• Pose open-ended questions about everyday math (e.g. estimating quantities)
• Discuss multiple ways to solve practical math problems
• Read books that incorporate mathematical ideas
• Use math apps that emphasize conceptual understanding

8. Developing number sense and flexible thinking is crucial for mathematical success

The most important factor in school success is what they call "opportunity to learn."

Number sense goes beyond memorizing facts. Students with strong number sense can work flexibly with numbers, understanding relationships between quantities and using efficient mental math strategies. This foundation is crucial for higher-level math success. Students who rely solely on memorized procedures often struggle as math becomes more complex.

Flexible thinking enables problem-solving. When students can approach problems from multiple angles and represent ideas in different ways, they're better equipped to tackle novel situations. This flexibility is more valuable than speed in executing standard procedures.

Ways to develop number sense and flexible thinking:

• Encourage mental math and estimation
• Discuss multiple strategies for solving problems
• Use visual models to represent mathematical relationships
• Play games involving strategic number use
• Emphasize conceptual understanding over rote memorization
• Practice decomposing and recomposing numbers

9. Mistakes and struggle are valuable for brain growth and deeper learning

When students make a mistake in math, their brain grows, synapses fire, and connections are made.

Productive struggle leads to deeper understanding. When students grapple with challenging problems, even if they don't immediately succeed, they are developing important problem-solving skills and mathematical mindsets. This struggle is where true learning occurs, as students make connections and develop new strategies.

A classroom culture that values mistakes promotes risk-taking. When teachers and students view mistakes as learning opportunities rather than failures, it creates a more positive and engaging math environment. Students become more willing to tackle challenging problems and share their thinking.

Ways to promote productive struggle:

• Provide challenging, open-ended problems
• Encourage multiple solution strategies
• Discuss and analyze incorrect answers for insight
• Praise effort and perseverance, not just correct answers
• Model making and learning from mistakes
• Use growth mindset language about mathematical ability

10. A growth mindset approach improves math achievement and persistence

All students can achieve at the highest levels of math in school if given the right opportunities and support.

Beliefs about mathematical ability impact achievement. Students who believe that math ability is fixed and unchangeable are more likely to give up when faced with challenges. In contrast, those with a growth mindset - who believe that abilities can be developed through effort and learning - show greater persistence and achievement.

Teacher and parent messages shape mindsets. The way adults talk about math ability has a powerful impact on children's beliefs. Praising effort and strategy use rather than innate "smartness" promotes a growth mindset. It's also important to challenge societal messages about who can be "good at math."

Strategies to promote a mathematical growth mindset:

• Emphasize that everyone can improve their math abilities
• Praise effort, strategies, and improvement, not speed or innate ability
• Provide challenging work and support productive struggle
• Discuss how the brain grows and changes with learning
• Share stories of mathematicians who overcame obstacles
• Avoid labels like "math person" or "not a math person"

Last updated: October 12, 2024