Mathematical Mindsets: The Highlights {Part 3}

This book I would say has changed my thoughts on math, teaching, and teaching math more than any other I've read in my seven year career. I will recommend it and link it forever. I will have to post my highlighted notes from it in several posts because no one would ever scroll through all of it otherwise! There is just so much to process and that I will need to read over and over again- so many opportunities for growth and change!

It's only $10.71 for the paperback and $7.99 for the Kindle version. You NEED this book. But until you get your own, this should be enough to make you want more.

See Part 1{here}and Part 2 {here}

Mathematical Mindsets: Unleashing Students' Potential through Creative Math, Inspiring Messages and Innovative Teaching
Jo Boaler

Chapter 5: Rich Mathematical Tasks

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. Studies have shown that the teacher has a greater impact on student learning than any other variable (Darling-Hammond, 2000).

This is intrinsically interesting, but it's also true that most people I meet, even high-level mathematics users, have never realized numbers can be so open and number problems can be solved in so many ways. When this realization is combined with visual insights into the mathematical ways of working, engagement is intensified.

I have learned through this that people are fascinated by flexibility and openness in mathematics. Mathematics is a subject that allows for precise thinking, but when that precise thinking is combined with creativity, flexibility, and multiplicity of ideas, the mathematics comes alive for people.

Teachers can create such mathematical excitement in classrooms, with any task, by asking students for the different ways they see and can solve tasks and by encouraging discussion of different ways of seeing problems.

They tried out ideas with each other, many of which were incorrect but helpful in ultimately forming a pathway to the solution.

Important observations that reveal opportunities to improve the engagement of all students:

  • The task is challenging but accessible .
  • The boys saw the task as a puzzle
  • The visual thinking about the growth of the task gave the boys understanding of the way the pattern grew
  • They had all developed their own way of seeing the pattern growth
  • The classroom had been set up to encourage students to propose ideas without being afraid of making mistakes
  • We had taught the students to respect each other's thinking
  • The students were using their own ideas,
  • The boys were working together
  • The boys were working heterogeneously.

When we don't ask students to think visually, we miss an incredible opportunity to increase their understanding.

Additionally, students did not think they were finding a standard answer for us; they thought they were exploring methods and using their own ideas and thoughts, which included their own ways of seeing mathematical growth.

The researchers found that when students were given problems to solve, and they did not know methods to solve them, but they were given opportunity to explore the problems, they became curious, and their brains were primed to learn new methods, so that when teachers taught the methods, students paid greater attention to them and were more motivated to learn them.

The teacher taught them the methods when they were needed, rather than the usual approach of teaching a method that students then practiced.

When students are asked to think intuitively, many good things happen. First, they stop thinking narrowly about single methods and consider mathematics more broadly. Second, they realize they have to use their own minds—thinking, sense making, and reasoning. They stop thinking their task is just to repeat methods, and they realize their task is to think about the appropriateness of different methods. And third, as the Schwartz and Bransford research study showed, their brains become primed to learn new methods (Schwartz & Bransford, 1998).

When teachers are designers, creating and adapting tasks, they are the most powerful teachers they can be.

Making math tasks richer:
1. Can You Open the Task to Encourage Multiple Methods, Pathways, and Representations?
2. Can You Make It an Inquiry Task?
When students think their role is not to reproduce a method but to come up with an idea, everything changes (Duckworth, 1991).
The mathematics is more complex and exciting because students are using their ideas and thoughts.
3. Can You Ask the Problem Before Teaching the Method?
4. Can You Add a Visual Component?
5. Can You Make It Low Floor and High Ceiling?
When students are invited to ask a harder question, they often light up, totally engaged by the opportunity to use their own thinking and creativity.
6. Can You Add the Requirement to Convince and Reason?

In every math conversation, students were asked to reason, explaining why they had chosen particular methods and why they made sense. This opened up mathematical pathways and allowed students who had not understood to both gain understanding and ask questions, adding to the understanding of the original student.

She explains that there are three levels of being convincing (Boaler & Humphreys, 2005):

  • Convince yourself 
  • Convince a friend 
  • Convince a skeptic 
It is fairly easy to convince yourself or a friend, but you need high levels of reasoning to convince a skeptic. Cathy tells her students that they need to be skeptics, pushing other students to always give full and convincing reasons.

When I ask students to play the role of being the skeptic, I explain that they need to demand to be fully convinced. Students really enjoy challenging each other for convincing reasons, and this helps them learn mathematical reasoning and proof.

Open up the task so that there are multiple methods, pathways, and representations. Include inquiry opportunities. Ask the problem before teaching the method. Add a visual component and ask students how they see the mathematics. Extend the task to make it lower floor and higher ceiling. Ask students to convince and reason; be skeptical.

Chapter 6: Mathematics and the Path to Equity

When we have gifted programs in schools we tell students that some of the students are genetically different; this message is not only very damaging but also incorrect.

Some people who have excelled in math choose not to be proud of the hard work and struggle they went through; they prefer to think they were born with a gift. There are many problems with this idea, one being that students who are successful through hard work often think that they are imposters because their achievement was not effortless.

The researchers went on to study the factors in the students' environment that led to different feelings of belonging, and they found that two factors worked against feelings of belonging. One was the message that math ability is a fixed trait; the other was the idea that women have less ability than men. These ideas shaped women's, but not men's, sense of belonging in math. The women's lowered sense of belonging meant that they pursued fewer math courses and received lower grades. Women who received the message that math ability is learned were protected from negative stereotypes—they maintained a high sense of belonging in math and remained intent on pursuing mathematics in the future.

We need all teachers to believe in all students, to reject the idea of some students being suitable for higher-level math and others not, and to work to make higher-level math available to all students, whatever their prior achievement, skin color, or gender.

Some teachers believe that some students cannot achieve at high levels of high school because they live in poverty or because of their previous preparation. In Chapter One I gave an example of high school teachers who made this argument to their school board, but teachers such as those at Life Academy are proving this wrong every day, through teaching high-level mathematics and positive messages to all students.

This is unfortunate, as we know that students who are advanced in math from an early age are more likely to drop math when they get the opportunity and achieve at lower levels.

Making math more equitable:
1. Offer all students high-level content
2. Work to change ideas about who can achieve in mathematics
The studies also show, encouragingly, that students who have a growth mindset are able to shrug off stereotyped messages and continue to success; this speaks again to the huge need for students, and teachers, to develop growth mindset beliefs about their own subjects and transmit growth mindset messages to students.
3. Encourage students to think deeply about mathematics
Unfortunately, the procedural nature of mathematics teaching in many classes means that deep understanding is often not available, and when girls cannot gain deep understanding they underachieve, turn away from mathematics, and often develop anxiety. Girls have much higher levels of anxiety about mathematics than boys do (Organisation for Economic Co-operation and Development [OECD], 2015), and the unavailability of deep understanding is one main reason for this (Boaler, 2014a).
4. Teach students to work together
When the Chinese American students found mathematics difficult, they were supported—first by knowing that everyone was struggling and then by working together to solve problems.
5. Give girls and students of color additional encouragement to learn math and science
The researchers found that the levels of anxiety held by women elementary teachers predicted the achievement of the girls in their classes, but not the boys (Beilock et al., 2009).
Researchers found that when mothers told their daughters “I was no good at math in school” their daughter's achievement immediately went down (Eccles & Jacobs, 1986). Teachers need to replace sympathetic messages such as “Don't worry, math isn't your thing” with positive messages such as “You can do this, I believe in you, math is all about effort and hard work.” Subsequent experiments showed that women underachieved when they simply marked their gender in a box before taking the test, compared to those who did not have to do that. Role models are extremely important to students—and one of the reasons it is so important to diversify the teaching force.
6. Eliminate (or at least change the nature of) homework
PISA, the international assessment group, with a data set of 13 million students, recently made a major announcement. After studying the relationships among homework, achievement, and equity, they announced that homework perpetuates inequities in education (Program for International Student Assessment [PISA], 2015).

Additionally, they questioned whether homework has any academic value at all, as it did not seem to raise achievement for students. This is not an isolated finding; academic research has consistently found homework to either negatively affect or not affect achievement. Baker and LeTendre (2005), for example, compared standardized math scores across different countries and found no positive link between frequency of math homework and students' math achievement. 

Mikki (2006) found that countries that gave more math homework had lower overall test scores than those that gave less math homework (Mikki, 2006). Kitsantas, Cheema, and Ware (2011) examined 5,000 15-and 16-year-olds across different income levels and ethnic backgrounds and also found that the more time students spent on math homework, the lower their math achievement across all ethnic groups.

When we assign homework to students, we provide barriers to the students who most need our support. This fact, alone, makes homework indefensible to me.

It is unfair and unwise to give students difficult problems to do when they are tired, sometimes even exhausted, at the end of the day. I wonder if teachers who set homework think that children have afternoon hours to complete it, with a doting parent who does not work on hand. If they do not think this, then I do not understand why they feel they can dictate how children should spend family time in the evenings.

The value of most math homework across the United States is low, and the harm is significant.

Homework should be given only if the homework task is worthwhile and draws upon the opportunity for reflection or active investigation around the home.

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