This blog post is a quick summary of the book “Math-ish” written by Jo Boaler.

A New Mathematical Relationship

The book starts with the author having a dinner with a CEO who was skeptical about any need to change the way mathematics is taught. Through a visual illustration, the author convinces the CEO and the rest that math education needs to embrace diversity in people and diversity in the approaches. These two powerful forces can tear down the problems created by narrow mathematics taught to most students around the world.

In the world of narrow mathematics, questions have one valued method, and one answer. They are always numerical, and they do not involve visuals, objects, movements or creativity. Most people have only ever experienced narrow mathematics, which is we have a country of widespread mathematics failure and anxiety.

Another issue that the author highlights about many people having negative relationship with the subject is that, it is often overly tested subject in any school.

It is often used to rank students, and by extension, to measure their worth of people. Students often do not even think about math for its own sake; they can only think about how well they are doing in the subject. To make it worse, testing usually consists of cold, narrow questions taken at speed.

The other two characters that have created problems in mathematical education are misrepresentation of math as a set of procedures and presence of “math-brain”, a somewhat mysterious trait only present in a few individuals.

For centuries, many believed that some people were born with a “math brain” and could learn math to high levels, while others could not. Often these ideas were combined with sexist, racist, and other discriminatory ideas about who had these special “gifts.” But the past ten years or so have shown, quite definitively, that there is no such thing as a math brain, and all brains are constantly developing, connecting, and changing. This is supported by neuroscientific evidence revealing incredible brain growth after short interventions. It has also been demonstrated by people whose early years of school were difficult, often being labeled as needing severe special education support, but who went on to achieve the highest mathematical levels— including a doctorate in applied mathematics from Oxford University.

The author cites the research of a neuroscientist Lang Chen who has shown that attitudes towards math correlates to math achievement. This means that it makes more sense to encourage people to have a positive mindset and enjoy mathematics. That alone can bring a mighty change in the performance levels of the students.

With this as the primer to what’s come, the subsequent chapters of the book delve in to the details of this new model of imparting mathematical education to students at all levels.

Learning to Learn

The author cites the research study carried out by John Hattie on 300 million students, that highlights metacognition as having the maximum effect size in the study of relationship between different approaches in education and student achievement.

A person who has learned metacognitive strategies is likely to be inquisitive and curious, they are eager to learn, and they appreciate diverse viewpoints. If they are stuck in a problem, they may circle back and think about what they know and need to know, or they may choose from other different strategies they have learned. Importantly, they enjoy the process of problem-solving and learning. This complex combination of high-level problem-solving, mindset, and planning that occurs when we are metacognitive takes place in the anterior prefrontal cortex of our brains. When people learn to be metacognitive, they not only improve their problem-solving, they also develop greater prosocial behavior, become better communicators, develop more empathy, and learn greater executive control. Some people learn to engage in these different ways, invoking a complex combination of mindset and higher-order thinking through their lives, and they are more successful people because of it.

The author lays out eight mathematical strategies to improve metacognition:

1. Take a Step Back
2. Draw the Problem
3. Find a New Approach
4. Reflect on “Why”?
5. Simplify
6. Conjecture
7. Become a Skeptic
8. Try a Smaller case

The author suggests that Journaling might be a great option to encourage Metacognition

I am a big fan of giving learners journals in which they reflect on their mathematics learning journeys. These are not the exercise or workbooks typically given out in math class, where students record answers; they are instead open spaces for free thought and reflection. It is not only students who benefit from having journals in which to set out their thinking and their reflections; we all benefit from having spaces for our reflective thoughts. I do not go anywhere without my own journal to record my thoughts, ideas, and plans. I prefer journals with pages that are blank or that have lightly dotted squares, so that students can think outside the lines, literally. We give students journals in our math camps and invite them to write down any useful ideas about mathematics or their own learning. We also give students time at the start of class to decorate their journals, so they feel ownership of them.

Valuing Struggle

The author says that one of the key ways to understand and become better at math is to embrace the struggle that comes with tackling a subject like math. Focusing on the mistakes and thinking of another way to solve the same question, is a doorway to expanding your brain to be plastic and become flexible in thinking about the problem from many ways.

The author mentions that she introduces her undergrad class at Stanford with a video of a classroom in Japan and the various interactions that take place during the class. Through the video, the author sends a message across to the class that 1) It is important to be investigative when solving problems, 2) It is important to have visual aids or cues in solving problems 3) It is important to collaborate and share mistakes

We should embrace or even seek out opportunities to do work that pushes us to the edge of our understandings, as it is on that edge that the greatest knowledge can be discovered; it is where creativity is found and important discoveries can be made. Often, when we venture out onto the edge of places where we are unsure, lack knowledge, or have uncertainty, that is where we achieve the greatest accomplishments. We do not achieve much in life by playing it safe or giving in to our inner negative voices and fears.

Brain is much more plastic than most people can think. Work on a task a few minutes each day for 6 weeks can change the structure of the brain. Now Imagine what you can accomplish in a year working on something few minutes each day

Mathematics in the World

The author starts off by mentioning her goal in writing this book

My goal in writing this book is to share the idea of math-ish, and to celebrate mathematical diversity. Math-ish, and the concept of “ish,” as you will read later in the chapter, is a diverse approach to understanding mathematics as it exists in everyday life for many different kinds of people. The concept of mathematical diversity encompasses the value of cultural diversity and difference in people, and diversity in approaching mathematics, appreciating different ways of seeing and thinking.

The author says it is crucially important to teach students a few key skills rather than reaching everything at a high level. The three skills are considered as crucial skills are

Number sense

Thanks to this book, I came to know about Cuisenaire rods that can be used as a tactile way to teach addition, subtraction, multiplication and division

Mathish

This is probably the most important skill that I never learnt in my school nor in my engineering education and probably not in my masters. The ability to estimate is so critically important in our lives and somehow this is never emphasized in the education. The author is spot on when she says

Numbers are everywhere in the world and we all use them, in some form, every day of our lives. But there is something noteworthy about our everyday use of numbers that differs from the ways we use and learn numbers in school. When we use numbers in our lives and workplaces, they are nearly always imprecise estimates, what I call ish numbers. To some people the idea of ish numbers is heresy, as they believe that numbers have to be accurate, precise, and correct at all times. But ish numbers turn out to be the numbers we most need in our lives, and I believe they could transform people’s approach to mathematics if they were present in their learning journeys.

The ability to judge whether an answer to a calculation is reasonable may be the most valuable mathematical ability that any student or adult person can develop, yet most students are missing it. I am not surprised, as mathematics learning worldwide focuses on precision and accuracy, and estimation and ish-ness are neglected.

Also why to use a new word like “mathish” instead of the word “estimate” ?

Teachers can use ish when talking with students who are nearly there, but not quite there. When we bring ish into our lives we are more protected from the dangers of perfectionism— a damaging mindset, and from binary thinking. Some people have asked me how asking students to “ish” numbers is different from asking them to estimate. It is a language change, but it is an important one, and this is why. When we ask students to estimate they think they are being asked to perform another mathematical method. But when we ask them to ish numbers they feel free and they are more willing to share their ideas— at the same time as developing number sense.

Data Literacy

This is a feature that we are seeing in schools thanks to democratization of data and tools. Teachers all over the world are bringing in data literacy based lessons at a much younger age. If a school or teacher isn’t incorporating these, the arguments made by the author makes it a compelling case to start imparting visualization skills, linear relationship modeling skills and general data awareness skills to the students

Mathematics as Visual Experience

The author talks about the importance of having mental representations in understanding new mathematical concepts and retaining old ones. In this context, the author cites Anders Ericsson, a world expert on expertise who says

The most important quality of deliberate practice is the opportunity it provides to develop mental representations. A second important quality of deliberate practice is the opportunity it gives students to struggle. You don’t build mental representations by thinking about something; you do by trying to do something, failing, revising and trying again, over and over

The author also quotes a research finding that shows the high-achieving academics who work in non-mathematical fields use visual brain areas much more than language areas.

Two of the brain pathways, which focus on visuals, are at the back of our heads. When we encounter a problem in numbers and see a visual representation, or a well-worded description, connections are made between brain regions. Researchers have shown that students achieve more highly when they are given mathematical work that involves both numbers and visuals.

The author several strategies to incorporate visual thinking in the mathematical curriculum

• Groupitizing
• Use of Cusinaire rods that give students to have mental representations of numbers and operations
• Number Path Visual
• Turn the problems in to something relating to the world
• Encourage students to develop mental representations by inviting them to think not only visually but also physically. Make students think about multiplication physically, visually and numerically
• Using examples to transition between numbers and algebra

The Beauty of Mathematical Concepts and Connections

The author highlights the importance of developing number sense among students.

when students learn to count, they are learning a method, but that should lead into an understanding of the concept of a number. When they learn the method of counting on, it should lead into a concept of a sum, and when they learn the method of addition, it should lead into the concept of a product. Learning concepts involves thinking deeply, considering, for example, What is a number? How can it be broken apart in different ways to make other numbers? How can it be represented visually? Where do we see numbers in the world? Some students never learn to think conceptually because the teaching they experience is all about rules and methods.

The author mentions that one of the main problems with rule-based approach is the understanding cannot be compressed. Only concepts can be compressed

Mathematics is amazingly compressible: you may struggle a long time, step by step, to work through the same process or idea from several approaches. But once you really understand it and have the mental perspective to see it as a whole, there is often a tremendous mental compression. You can file it away, recall it quickly and completely when you need it, and use it as just one step in some other mental process. The insight that goes with this compression is one of the real joys of mathematics.

The book mentions the importance of learning the key ideas as a connected concept graph. When students can dive deeply in to the mathematical concepts, they learn the same mathematics, but instead of learning disconnected methods piece by piece, they learn a set of connected ideas and methods through rich tasks.

Here is Nueroscientist Jeff Hawkin’s proposal

He argues that the brain arranges all knowledge using reference frames— these are located in our brains, as big cities might be located on a map. 9 As mathematicians work, we need reference frames to know where we are and what we need to do to move from one area of knowledge to another. Often these reference frames have visuals that we can see or imagine in our minds. Hawkins gives the example of DNA. If you have studied genetics, when someone talks about DNA, your brain is likely to pull up a visual of a double helix that can unzip. You have never seen a DNA molecule, but your brain has made a reference frame for this area of knowledge, which helps with organization. If people are learning effectively, they are making reference frames. They are not just storing facts, one on top of another; they are relating their knowledge to bigger, conceptual reference frames. Hopefully, these reference frames have mental representations to ground them. Our recommendation is to help students learn a set of big ideas deeply and well, so that these big ideas serve as reference frames that can organize all the students’ knowledge.

In order to crystallize all of the information and concepts, the author encourages students to journal, make sketch notes, jotting down visuals

Diversity in Practice and Feedback

The author talks one of the most important aspect in learning and understanding mathematics - deliberate practice

Deliberate practice is the engagement with meaningful ideas, through which students develop representational models, and it includes a clear feedback loop to provide opportunities for improvement.

The definition as it stands, all the three elements of deliberate practice are missing in today’s math education. The author gives an example of two schools where contrasting math practices are followed. First one has procedural mind numbing math, Second one teaches math through physical, tactile, visual and numeric ways. Be it understanding the concept of locus or understanding negative numbers, the use of conceptual knowledge seems to have trumped over rote math and procedural math

The author also highlights the importance of learning math concepts using contrasting examples

So, what are the aspects that one should keep in mind when applying the concepts of Anders Ericsson to math education

Characteristics of Effective Practice

1. Application of methods: Problems should ask people to use methods in new and different situations.
2. Consideration of contrasting cases
3. Focus on concepts and big ideas, not small methods
4. Development of representational models that include visual or physical referents
5. Nonstandard examples and representations
6. Connections that people can see and learn; connections between mathematical ideas and between mathematics and the world

The author focuses on the way one can integrate feedback in to the math curriculum.

A New Mathematical Future

The author summarizes all the points mentioned in the book in the final chapter with a few visuals

The author mentions of an unusual mathematical competition designed by Sol Farfunkel

Sol has designed a competition that assesses mathematical modeling. Over a four-day period in each year of his competition, approximately 80,000 students work in groups of up to three people on difficult applied and diverse mathematical problems. Examples of problems include analyzing renewable energy in different states, examining trends in global languages, and planning an optical helicopter-search pattern. When students engage in these problems, they work with mathematical diversity and math-ish— drawing from different areas of mathematics, thinking in different ways, collaborating with each other, and building on each other’s ideas. Impressively, 43 percent of participants are women, and 43 percent of winners are women. At first the competition was designed for college students, but in the third year it was won by a team of high school students— the organizers had not even known that a high school had entered the competition. Ever since that time, this powerful mathematics experience has attracted and welcomed more and more high school teams.

The author investigated the reasons behind the mathematical diversity in the winners. Her findings are summarized in this visual

The author reflects on her school teacher Mrs.Marshall and credits her for showing math in a different life

What is interesting to me now, when I reflect on the experience I had as a seventeen- and eighteen-year-old student, is that the teacher transformed mathematics for me by changing two aspects of the mathematics learning experience: she invited students to talk about ideas, and she taught new methods after we had discussed situations and found a need for them, a practice that I have highlighted through several examples in this book.

Takeaway

Mathematics curriculum is broken across many countries. Step in to any math class and you will mostly see only a handful of students showing interest. Most of the students are then lead to private tuition classes after school to tide over the inefficient learnings from the school. This is a systemic issue and Jo Boaler offers practical strategies to change the math education so that it becomes more diverse, mathish and application oriented. You will like this book if you wonder about what can be done as citizens, parents, teachers so that we can impart effective math education to one and all.