More Boaler!

I am so glad I discovered Jo Boaler’s work.  She is inspiring me to rethink the way I teach math.  For my Inquiry class, we had to pick a peer-reviewed journal article to summarize.  I immediately searched for Boaler.  I found one of her most famous articles, and below I share my summary of her research.

Boaler, J. (2008). Promoting ‘relational equity’ and high mathematics achievement through an innovative mixed-ability approach. British Educational Research Journal, 34 (2), 167-194.

Summary of Research

This study followed students placed in mixed ability collaborative groups in math and compared them with students in traditional math classes over four years. Through lesson observations and student interviews and assessments, Boaler found that the students placed in the heterogeneous collaborative groups not only outperformed the students in traditional math classes in mathematics achievement but also showed increased relational equity, treating each other with more respect and offering each other more support.

Design of study:

Quantitative and qualitative data on 700 students were collected over four years at three California high schools. Two high schools used traditional teaching methods, which incorporated 21% lecture, 15% teacher questioning, 48% individual work, and 11% group work in ability-grouped math classes. The third school used reform methods, with 4% lecture, 9% teacher questioning, 72% group work, and 9% student presentations in heterogeneously grouped math classes.

Data Collection:

Over 600 hours of lessons at all three schools were observed and analyzed. Sixty students were interviewed every year, sampling from high and low achievers and different cultural and ethnic groups. All students were given questionnaires in the first three years of the study asking about their experiences and perceptions of math. All students were assessed for mathematics understanding, and selected focus groups were given applied mathematics assessments that were videotaped.

Emergent Themes:

  1. Students who work in heterogeneous groups learn to take responsibility for their own learning and the learning of others in the group resulting in higher mathematics achievement for all members.
  2. Students who work in heterogeneous groups learn to value other people’s perspectives and respect diverse cultures and backgrounds, which has positive implications for their future participation in a global economy.
  3. Students who work in heterogeneous groups show improved communication and collaboration, as well as enhanced critical thinking and problem solving.
  4. Teachers need to conscientiously support and model group work in order for it to be successful.


Despite starting out at lower levels, students at the reform school outperformed students at the traditional school within two years. More students (41% vs. 27%) at the reform school went on to advanced math classes by the fourth year of the study. Achievement differences between ethnic groups were reduced. The students at the reform school showed increased relational equity: helping others learn, taking responsibility (59% vs. 5%), and respecting other’s viewpoints.


Teachers require training to instruct students in the practices that make group work effective. Teachers need to help groups avoid unbalanced workloads, and make collaborative norms explicit. Teachers must also act as advocates for respect and responsibility, paying attention to how students are working together and not just to the mathematical outcomes.

 Implication of Study for Stakeholders:

The traditional teaching method in which teachers dispense information and individual students process it is not the most effective way to improve academic outcomes, and it does not address social justice, social and emotional learning, or citizenship goals at all. Grouping students heterogeneously by race, ethnic group, gender, and ability, and allowing them to work together to process a given topic improves outcomes in all of these areas, as well as engaging students in the development of twenty-first century skills such as collaboration, critical thinking, and communication. Teachers need support and training to enact these radical changes to their teaching methods so that they are implemented effectively in their classrooms.

Math in Life, Nature, & Work

The final lesson in Jo Boaler’s “How to Learn Math (for Students)”.

Math is all about finding patterns.  One of the most famous patterns was discovered by Fibonacci.  He found that if we start with 1, 1, and then add the two previous numbers, we get a pattern that looks like this: 1, 1, 2, 3, 5, 8, 13, 21,…. If this pattern is drawn as squares, it creates a spiral that is found in art and nature.

Snowflakes have a pattern, too.  Even though every snowflake is different, every snowflake has something in common: they all have six points.  This is because water freezes in a hexagon, so snowflakes are also hexagonal.

Animals intuitively use math.  Take the spider:  its web is created using a logarithmic pattern to stabilize it, and then the spider completes the web using an arithmetic pattern.  It would take an engineer quite a while to figure that out, but a spider does it intuitively. Dolphins use math, too.  They use it to echolocate.

Athletes use math intuitively as well. And dancers are extremely good at using physics and math to create complex patterns. Jugglers use tons of math!

The work world uses amazing amounts of math.  It’s not about being fast (we have computers for that) but problem solving and thinking about numbers creatively. Math in the real world is hands on and collaborative.

The modern world gives us math games that are really fun.  Here are some of Jo’s favorites: Wuzzit Trouble, Motion Math, DragonBox, and Mathbreakers.

I really loved this course.  I’m going to save up the $125 to take her teacher-focused class.

The world is mathematical!

Number Patterns and Representations

This is the fifth lesson in Jo Boaler’s “How to Learn Math (for Students)”.

It is up to us to make sense of math.  One way is by using intuition, which is a general sense or feeling about what works. According to Sebastian Thrun, founder of Udacity, “Every time you see a math problem it is useful to stop and think intuitively, what do I think is going on here?”

A great aid to intuitive thinking is visual representations through drawing.  That takes the math out of the realm of formulas and algorithms and puts in into a concrete context.

There are “Big Ideas” in math that the formulas and algorithms are about exploring and using.  However, sometimes in math class, those big ideas get lost, and consequently, students get confused.  For example, the Big Idea of Pi:  In every circle and every sphere the circumference is always a little bit more than three times the diameter. If students can learn the big ideas and hold onto them, they will find math really clear and easy to use.

In other words, formulas are only useful if you understand the big idea that they are illustrating, so students should always start by trying to understand the big idea.  Drawing a picture helps a lot.  Keeping in mind that math is about relationships between quantities helps too.  So asking questions about how the quantities relate (like diameter and circumference) can help facilitate understanding of the big idea (pi is the same for all circles).