Here's 5 books that have been influential to me as
someone concerned with math education.
If you read no other book, read this one:
The Mathematical Experience by Reuben Hersh and Philip Davis. If you've ever wondered what a mathematician is and what they do, this will help answer a few questions. And regardless of who you are, you'll end up with more questions about these topics than you've got answers.
The book is nicely organized into inner and outer issues. The outer issues tackle what mathematics and mathematicians can and do mean to the rest of the world. In some ways it is harsh, often depicting mathematicians as ineffectual loners, but at the same time the portrayal is honest. There are no cheap shots. And shortcomings are dealt with on their own terms, not merely in caricature. The inner issues are fantastic because they present not only tidbits of mathematics that is fascinating and which non-specialists rarely see, but these same topics bring up central philosophical difficulties in mathematics. One can learn a lot about the discipline just by coming to terms with why people might find these issues troublesome and worth the bother.
I found this book randomly as an undergrad and loved it, not knowing that one of the authors worked where I went to school. Growing up a reader one often depersonalizes the voice behind the words on the page. Knowing Reuben Hersh and thinking about him walking the campus just as I did brought his work home to me as I could not have expected.
My number two book is also by Hersh and Davis, Descartes Dream. The title refers to the philosopher's dream of understanding the entire cosmos through the lens of mathematics. The book examines this notion, where it came from, the form it has taken in modern society from technollogically based warfare to computer dating to standardized testing. As the previous book does, this book remains relevant (though in some ways outdated) because its many criticisms are honest and not overstated for shock value. I also probably enjoy the voice of a critical insider.
The Visual Display of Quantitative Information by Edward Tufte is next because purists tend to leave pictures out of math literature and pedagogy(and other academic language). Even when they are included it is somehow perceived as a crutch, a helpful hint for amateurs but the serious stuff is done with symbols and words. I believe a lot is missed here, and that this view is a vestige of outdated formalist notions of mathematics (and more genrally academics). Moreover, the data one encounters on a daily basis is more often than not presented graphically, so an understanding of the aesthetics involved is a necessary part of being able to critically analyse one's world.
Edward Tufte has made his name by promoting a specific style or language for traffiking in pictures and graphs. What makes a good diagram nice to look at is what makes the information it is attempting to convey transparent to the observer. I appreciate his style and his books show great examples of his principles both by displaying famous failures (he directly attributes the Challenger disaster to poor visual display and analysis of important data) and masterpieces of design. In making graphs for the proof project I have tried to follow his lead.
Maggie Lampert's Teaching Problems and the Problems of Teaching is a small example of how a math classroom can be run like a mathematical community, regardless of age or content level. Lampert describes a year teaching 5th grade math in a way that's unfamilliar to most.
The class is organized around somewhat large-scale problems that are presented not as a means for ritualized practice of algorithms, but as the central issue for discussion. Emphasis is not placed on the answer to the problem, and the teacher's expository role is mostly limited to directing students comments toward fruitful ends. The students work alone, in small groups, and have full-class discussions about their methods and solutions.
This format allows students not only to develop techniques for solving problems, but in making their reasoning explicit for others and evaluating others explanations. These are the central tasks of a mathematical professional, and here we see that it is not impossible for 5th graders.
The book works as proof of concept or an eye opener. The reader is left fascinated by what they have seen but with doubts as to how the math classroom they've grown up with could ever look like this one.
Discussions of alternative pedagogies in math are often waylaid by the objection: 2+2=4. In math, unlike some other subjects, there are right answers and wrong answers. Math class should then give studnets the ability to get the answers right. Daniel Chazan's book, Beyond Formulas in Mathematics and Teaching: Dynamics of the High School Algebra Classroom attacks this objection where it is least questioned: high school algebra.
Basically, the book follows this line of thinking. Once you ask 'What is 2+2?' the only answer is 4. The key to understanding what's missing from high school algebra is why we're asking that over and over again. That is not a simple matter, or one that is closed to debate. Once you're willing to see that as important, math is no longer 'cut and dried'. The book presents ways to see this ignored ambiguity in math curricula as well as what good can come out of seeing it.
Empowering Education: Critical Teaching for Social Change by Ira Shor is not specifically about math ed. Books written about the intersection of social justice and math ed tend to miss the points the above authors would make about math pedagogy. They focus on professional underepresentation and test performance. Shor gives specific strategies for making student and community participation the basis of activity in the classroom. It is an excellent companion to Chazan, the former giving the 'why' to his 'what'.
If you read no other book, read this one:
The Mathematical Experience by Reuben Hersh and Philip Davis. If you've ever wondered what a mathematician is and what they do, this will help answer a few questions. And regardless of who you are, you'll end up with more questions about these topics than you've got answers.
The book is nicely organized into inner and outer issues. The outer issues tackle what mathematics and mathematicians can and do mean to the rest of the world. In some ways it is harsh, often depicting mathematicians as ineffectual loners, but at the same time the portrayal is honest. There are no cheap shots. And shortcomings are dealt with on their own terms, not merely in caricature. The inner issues are fantastic because they present not only tidbits of mathematics that is fascinating and which non-specialists rarely see, but these same topics bring up central philosophical difficulties in mathematics. One can learn a lot about the discipline just by coming to terms with why people might find these issues troublesome and worth the bother.
I found this book randomly as an undergrad and loved it, not knowing that one of the authors worked where I went to school. Growing up a reader one often depersonalizes the voice behind the words on the page. Knowing Reuben Hersh and thinking about him walking the campus just as I did brought his work home to me as I could not have expected.
My number two book is also by Hersh and Davis, Descartes Dream. The title refers to the philosopher's dream of understanding the entire cosmos through the lens of mathematics. The book examines this notion, where it came from, the form it has taken in modern society from technollogically based warfare to computer dating to standardized testing. As the previous book does, this book remains relevant (though in some ways outdated) because its many criticisms are honest and not overstated for shock value. I also probably enjoy the voice of a critical insider.
The Visual Display of Quantitative Information by Edward Tufte is next because purists tend to leave pictures out of math literature and pedagogy(and other academic language). Even when they are included it is somehow perceived as a crutch, a helpful hint for amateurs but the serious stuff is done with symbols and words. I believe a lot is missed here, and that this view is a vestige of outdated formalist notions of mathematics (and more genrally academics). Moreover, the data one encounters on a daily basis is more often than not presented graphically, so an understanding of the aesthetics involved is a necessary part of being able to critically analyse one's world.
Edward Tufte has made his name by promoting a specific style or language for traffiking in pictures and graphs. What makes a good diagram nice to look at is what makes the information it is attempting to convey transparent to the observer. I appreciate his style and his books show great examples of his principles both by displaying famous failures (he directly attributes the Challenger disaster to poor visual display and analysis of important data) and masterpieces of design. In making graphs for the proof project I have tried to follow his lead.
Maggie Lampert's Teaching Problems and the Problems of Teaching is a small example of how a math classroom can be run like a mathematical community, regardless of age or content level. Lampert describes a year teaching 5th grade math in a way that's unfamilliar to most.
The class is organized around somewhat large-scale problems that are presented not as a means for ritualized practice of algorithms, but as the central issue for discussion. Emphasis is not placed on the answer to the problem, and the teacher's expository role is mostly limited to directing students comments toward fruitful ends. The students work alone, in small groups, and have full-class discussions about their methods and solutions.
This format allows students not only to develop techniques for solving problems, but in making their reasoning explicit for others and evaluating others explanations. These are the central tasks of a mathematical professional, and here we see that it is not impossible for 5th graders.
The book works as proof of concept or an eye opener. The reader is left fascinated by what they have seen but with doubts as to how the math classroom they've grown up with could ever look like this one.
Discussions of alternative pedagogies in math are often waylaid by the objection: 2+2=4. In math, unlike some other subjects, there are right answers and wrong answers. Math class should then give studnets the ability to get the answers right. Daniel Chazan's book, Beyond Formulas in Mathematics and Teaching: Dynamics of the High School Algebra Classroom attacks this objection where it is least questioned: high school algebra.
Basically, the book follows this line of thinking. Once you ask 'What is 2+2?' the only answer is 4. The key to understanding what's missing from high school algebra is why we're asking that over and over again. That is not a simple matter, or one that is closed to debate. Once you're willing to see that as important, math is no longer 'cut and dried'. The book presents ways to see this ignored ambiguity in math curricula as well as what good can come out of seeing it.
Empowering Education: Critical Teaching for Social Change by Ira Shor is not specifically about math ed. Books written about the intersection of social justice and math ed tend to miss the points the above authors would make about math pedagogy. They focus on professional underepresentation and test performance. Shor gives specific strategies for making student and community participation the basis of activity in the classroom. It is an excellent companion to Chazan, the former giving the 'why' to his 'what'.
Titles:
The Mathematical Experience
Descartes' Dream
The Visual Display of Quantitative Information
Teaching Problems and the Problem of Teaching
Beyond Formulas in Mathematics and Teaching
Empowering Education: Critical Teaching for Social Change
