Boaler is mostly critical as she scrutinizes various aspects of the present state of affairs in K–12 mathematics education in America and Britain today. But she also offers a vision for the future, holding up as examples proven model approaches and providing ideas for teachers and parents to allow children to attain the best mathematics education possible.
Boaler begins by describing the importance and essence of mathematics and complains that the subject is not being taught in American schools. “Mathematics is now so critical to American citizens that some have labeled it the ‘new civil right’.” By “mathematics” she means not recall of particular methods, but reasoning and problem solving characterized by flexibility in new situations.
(I have to share an editing curiosity in my copy of the book. On the title page, the subtitle reads, “Helping Children Learn to Love Their Most Hated Subject — and Why It’s Important for America,” whereas on the book jacket the subtitle reads “Helping Children Learn to Love Their Least Favorite Subject — and Why It’s Important for America.”)
Most American adults hate mathematics because of their traumatic experience with it as children. “The math that millions of Americans experience in school is an impoverished version of the subject and it bears little resemblance to the mathematics of life or work or even the mathematics in which mathematicians engage.” Rutgers professor Diane Maclagan was asked, “What is the most difficult aspect of your life as a mathematician?” She replied, “Trying to prove theorems.” The interviewer then asked what the most fun aspect is. She replied, “Trying to prove theorems.” Students should work on long problems that give rise to interesting ideas, not short problems involving the “repetition of isolated procedures.”
Mathematics is a precise discipline, imbued with a precise notation. But precision does not necessitate drilled teaching methods. It is precision that frees up practitioners to focus on ideas. Mathematics education should do the same.
Mathematics is a collaborative subject, one in which questions are posed and contributions to a solution are made by many; thus, what transpires in the typical silent classroom is not mathematics. “All the mathematical methods and relationships that are now known and taught to schoolchildren started as questions, yet students so not see the questions. Instead, they are taught content that often appears as a long list of answers to questions that nobody has ever asked.”
Boaler briefly reviews the math wars, characterizing the traditional approach as the force-feeding of definitions followed by many practice exercises and the reform approach as the organic introduction of definitions via interesting problems. Some aspects of the traditional approach are good, such as its insistence on skills mastery. What must be avoided is passive learning: students copying down techniques and then practicing them. Lamentable is the perspective of many children, summarized by one girl: “In math you have to remember; in other subjects you have to think about it.”
Boaler rails against “learning without thought,” which she illustrates with a few math problems that stump many adults who simply try to apply a vaguely remembered mechanical procedure. One example math problem reads, “A woman is on a diet and goes into a shop to buy some turkey slices. She is given 3 slices which together weigh 1/3 of a pound, but her diet says that she is only allowed to eat 1/4 of a pound. How much of the three slices she bought can she eat while staying true to her diet?” Adults usually tried things like 1/3 x 1/4 or 1/4 x 3. In contrast, young children saw that 9 slices were a pound and proceeded to divide 9 slices by 4, either simply writing 9/4 slices or by drawing a picture of 9 slices and quartering the picture. [Note that however vague is the question “How much of the three slices…,” the answer children obtained is practical.]
A bit later in the chapter, Boaler also denounces so-called “real-world problems” given in traditional mathematics classes, claiming that most such problems are so artificial that to solve them students must ignore everything they know about the real world.
Presenting “a vision for a better future,” Boaler describes the interactive, reform mathematics classes at Railside, an urban high school in California. One example of an activity involved students studying a sequence of tile patterns and representing, in various ways, how the patterns grow. Students worked in groups, jumped from group to group, and shared at the chalkboard when a significant discovery was made. Comments from a couple of freshmen: “Back in middle school the only thing you worked on was your math skills. But here you work socially and you also try to learn to help people and get help… You can’t just be like ‘Oh, here’s the book. Look at the numbers and figure it out.’” “It’s not just one way to do it… It’s more interpretive… And then it’s like ‘Why does it work?’”
The author’s specialty is longitudinal studies, and one she presents involved Phoenix Park and Amber Hill high schools, demographically similar schools in England. Math classes at Phoenix Park are project-based, with students enjoying lots of freedom to choose complex problems that they would like to work on, whether they would like to work in groups or individually, and with the teacher introducing methods to students or groups only when the students have reached a point where the methods would be meaningful. Amber Hill uses the traditional approach. Phoenix Park students outperformed Amber Hill students on standardized tests, had a more positive view of mathematics, and used math more in their part-time jobs.
A follow-up study with the students when they were in their twenties found that Phoenix Park graduates were working in more professional jobs than Amber Hill graduates. Phoenix Park graduates considered their math education as having equipped them for problem-solving in various situations they encountered; Amber Hill graduate lamented that their math education had been pointless and disconnected from real life.
Boaler begins to address assessment by presenting evidence that high-stakes standardized testing lowers achievement. Most multiple-choice math tests are poor indicators of achievement, in part because they are discriminatory and in part they measure language ability as much as they do math ability. The only feedback is a percentile score, which spells doom for many who get low scores: controlled studies show that informing students that their socioeconomic group tends to perform lower than another is a self-fulfilling prophecy.
A better approach is “assessment for learning,” which involves three things: 1) enunciating clear content-focused learning goals, 2) allowing students to monitor their learning of these goals through individual and peer assessment, and 3) having comment-based rather than grade-based assessments, providing feedback on what has been done well and what still needs to be learned. The author provides evidence that assessment for learning is very effective, cheap, and can be done at a national level. (An example given is the system used in Queensland, Australia that combines moderated schoolwork and a basic skills test.)
The author next claims that tracking or grouping students according to attainment results in less learning for students of all attainment levels. This chapter focuses mainly on the damage done to students placed in lower tracks, lamenting the impact of lower expectations, lower self-esteem, loss of future opportunities (like entering college), and a lack of respect from students in higher tracks. Mixed classrooms produced better results. The ones that were described followed the reform model and included group work. Higher-attaining students learned more by explaining things to lower-attaining students, a culture of respect for different types of intelligence was established, and students achieved more later in life.
Next Boaler addresses gender differences, stating that girls are given short shrift in traditional mathematics classrooms. She notes that while boys tend to be content to learn a method and apply it many times, girls tend to want to know why something works and to make connections to other material. At Amber Hill School (which has traditional mathematics classrooms) boys outperformed girls. Girls fell behind even when they had entered the school near the top in mathematics. Both girls and boys achieved equally at Phoenix Park School (with reform mathematics classrooms) and at higher levels than the girls and boys at Amber Hill.
Boaler goes on to discuss brain differences between women and men (women are more wired for language and communication, women use both hemispheres at once). She points out that girls learn mathematics just as well, maybe better, than boys when given the opportunity. She deplores stereotypes and laments the paucity of women in mathematics faculty and other research positions.
Successful students don’t just know more — they think in different ways. For example, in arithmetic problems, they characteristically decompose and recompose numbers. Unsuccessful students, on the other hand, continue to rely on rudimentary methods like simple counting. “Often thought of as slow learners… they are learning a different mathematics,” a mathematics that is a more difficult subject.
As the years go by, and math problems become more complex, low achievers think they need to count more and more precisely. Successful intervention involves encouraging students to ask many questions, in turn asking students to justify their reasoning, highlighting the importance of using various mathematical representations, and cultivating a flexible use of numbers by asking students to mentally solve a given arithmetic problem and then having everyone share their methods. A summer intervention program using these principles resulted in huge attitude changes and achievement levels. (But most of the students became low achievers again soon after returning to a traditional classroom.) When asked what math teachers should do to make math class better, one of the participants responded, “Give harder problems.”
Boaler admirably devotes a chapter to helping parents encourage their children’s mathematical development. Providing mathematical settings is important. Building blocks, linking cubes, jigsaw puzzles, etc., help develop spatial reasoning, while pattern detection and creation can be practiced by manipulating Cuisenaire rods, pattern blocks, or a pile of nuts, bolts, washers, and colored tape. Mathematical puzzles are very important; many mathematicians cite puzzles posed by parents as the most important influence in their mathematical growth. Asking questions that probe children’s thinking prompts children to reflect on their own thinking. Finally, children should be introduced to problem-solving strategies (like Pólya’s), the use of which is one of the biggest factors that distinguishes high from low achievers.
A final chapter provides suggestions for parents who want to get more involved in the public schools and advocate for a more reform-based mathematics classroom. Parents are encouraged to talk to their children’s teachers, the principal, and the PTA. To prep parents for these interviews, Boaler provides conversation starters and a list of books and articles that support the claims she has made in this book.
Appendices provide solutions to the several math problems posed throughout the book, a list of mathematics curricula, and a list of math puzzle books.
Boaler is in some places provocative, in perhaps most places convincing. She covers several topics that are of critical importance to K–12 mathematics education. Some readers are already aware of the current conditions Boaler describes; others will be shocked by the picture she paints. Most readers will warm to her suggestions for improvement, even if they don’t agree with everything. This book is recommended reading for K–12 mathematics teachers and administrators, parents, and also for future mathematics teachers and those who educate them.
David A. Huckaby is an assistant professor of mathematics at Angelo State University.