Jo Boaler on Visual, Creative Mathematics Education

Boaler describes the beauty of mathematics as the possibility of seeing and solving problems in different ways. She argues that any area of maths can be made visual and that teaching it as numbers, one method, and one answer leaves out other ways of understanding.

Boaler discusses visual pathways and connecting representations through numbers, words, and physical models. Fridman relates visualization to physicists’ intuition and experiments; they consider how intuition can help generate an idea before formal proof, and why struggling with a problem can be valuable.

The conversation considers intuition, patterns, experiments, and computer-generated visualizations as ways to explore a problem. Boaler says visualization and color coding are underused in teaching and describes teachers responding strongly to seeing mathematical ideas represented visually.

Boaler says challenge can support learning when students believe they can succeed, and discusses the damaging idea that people are either born with a maths brain or not. She argues for offering multiple ways to access mathematics so students with different ways of thinking can engage.

Fridman reflects on the Soviet emphasis on excellence and extensive study of mathematics, geometry, science, and literature. Boaler contrasts this with students being told they may simply be unable to succeed, while noting that effort also needs good teaching. She describes Stanford summer camps where students remained engaged in extended mathematical activities.

Boaler identifies fifth grade and the middle-school years as important periods when students may start to lose confidence or decide that STEM subjects are not for them. She links this partly to changes in classroom expectations, including more grades and tests.

Boaler describes an English-class study in which a brief message expressing a teacher’s belief was associated with better student outcomes a year later. They discuss the need for teachers to mean their encouragement, the value of communicating high expectations, and how school systems may favor memorization over creative or deep thinking.

The conversation considers whether multidimensional teaching can support students who learn differently. Boaler discusses Maryam Mirzakhani’s visual mathematical work and her account of being told she could not do maths at school. Fridman and Boaler reflect on how a teacher’s actions can shape a student’s confidence and future.

Boaler discusses research linking parents’ maths anxiety with children’s achievement when parents help with homework, and says children can absorb negative messages about maths. She recommends that parents avoid transmitting anxiety and help children see maths as more than methods to memorize.

Boaler explains a proposed California maths framework organized around big ideas and connections rather than isolated standards. She gives unit squares as an example of how narrow standards can become narrow textbook questions, and describes using rich activities to help students explore broader mathematical concepts.

Boaler distinguishes a final grade from frequent grading during a course, which she says can create a performance-focused classroom. She describes rubrics, student self-assessment, and teacher notes as ways to show what students are learning and what they might work on next.

Fridman describes teachers who challenged and encouraged his particular interests. Boaler says teachers cannot know in advance which students will pursue an idea, and gives an example of a student extending a square-border problem to a pentagon. They discuss adapting encouragement to individual students.

Boaler explains an opening classroom activity in which students briefly view seven dots and describe how they grouped them. She says students can see the same dots in many ways, calls the ability “groupitizing,” and argues that valuing these different approaches can make maths engaging for learners of different ages.

Boaler argues that maths education should develop flexible, creative thinking rather than focus mainly on memorization and procedures. She says these human capacities matter in a world where computers can carry out many routine tasks.

Boaler describes teaching Stanford calculus students to collaborate after they had mostly experienced maths as individual and competitive. She says students learned to build on different approaches and did better on a shared problem. Fridman and Boaler discuss listening, setting aside ego, and avoiding social comparison.

Fridman recalls enjoying school textbooks, while Boaler notes that some students experience maths textbooks as a rigid sequence of exercises. They discuss how textbooks can be used, differences in how algebra and geometry are taught, and why Boaler prefers “maths” as a term that suggests a broader subject than calculation alone.

Boaler discusses her experience creating online courses after working with Sebastian Thrun and the response to a free course for maths teachers. She says online learning can reach more people but argues for interactive lessons with short presentations and activities rather than long stretches of talking.

Boaler describes YouCubed’s goal of sharing research-informed teaching ideas with teachers, students, and parents in accessible forms. She discusses the Week of Inspirational Maths, its K–8 activity books, and her wish for more teachers to know about and use the resources.

Boaler recommends testing understanding by trying problems rather than only rereading explanations. For homework, she suggests reflecting on the lesson’s big idea, difficulties, and real-world relevance. Fridman connects asking why to first-principles thinking and invention.

The speakers discuss how students’ questions can diminish over time and Boaler recalls a teacher’s dismissive response that stopped her asking questions in class. They contrast that experience with encouragement from another teacher, Mrs. Marshall, whose use of group discussion changed Boaler’s experience of maths.

Boaler says students need opportunities to work on rich, deep problems and describes teachers being surprised by how quickly an hour can pass while working on one. Fridman discusses focus as a skill that can be developed gradually. Boaler notes a survey finding that students said they would work on a maths problem for two minutes before giving up.

Boaler and Fridman discuss Grant Sanderson’s visual explanations, storytelling, and programmatic animations, as well as the value of making visualizations to teach. They connect mathematical thinking with finding patterns and discuss online courses. Boaler describes a six-session “How to Learn Maths” course for children and the research conducted on its effects.

Boaler encourages young people and adults to believe they can learn and to seek people who believe in them. Fridman reflects on the importance of mentors and supportive relationships, and Boaler discusses people who have been told they cannot do maths but later discovered other ways of thinking about it.

Boaler expresses hope for change in education while saying it often moves slowly. She discusses data science entering high-school curricula, programming and creativity, and the possibility of subject boundaries changing. The episode closes with Fridman thanking Boaler and the show’s closing message.