
The UCSB Center for mathematical inquiry has been funded the Educational Advancement Foundation to encourage development and research on inquirybased learning (IBL) in mathematics at all levels, Kindergarten through the graduate level. Many people wonder if IBL is a new movement and if it has a history success. In fact, IBL has a rich history and record of success. A few interesting examples are outlined here. Background. Throughout history educators in many fields in addition to mathematics have used IBL and it has taken many formats according to the subject and the styles of the instructors. It has long been recognized that involving students in research is a highly effective way to prepare them for careers, even in technical fields where one would think mastery of large amounts of information is the core curriculum. For example, in recent years many medical schools have started using case studies and problem solving as a central component of even their most basic courses, replacing the traditional lectures with student led inquiry, such as UC Berkeley’s threeyear 100% lecturefree inquirybased medical curriculum (after which students go to UCSF for their clinical work.) Perhaps the earliest welldocumented movement to use IBL in mathematics education in the US was the work of Warren Colburn (17931833) who wrote several arithmetic texts emphasizing student invention of computational procedures and mental arithmetic. Greater numeracy was a national need at the time with the country’s expanding economy, facilitated in part by the new decimal monetary system. The Boston Education Act of 1789 required that boys between 11 and 14 learn arithmetic through fractions. Later, in 1819 educator Samuel Goodrich, author of The Child’s Arithmetic, argued that teaching arithmetic by rote actually prevented children from understanding arithmetic and that they should discover rules by manipulating tangible objects^{1}. Discontent with arithmetic learning set the stage for Colburn’s successful books (a best selling 100,000 copies a year in the 1850’s). The preface to Colburn’s 1826 Edition of Arithmetic Upon the Inductive Method of Instruction describes his concept of IBL, 
Set a student to work on an addition problem without telling him what to do. He will discover what is to be done, and invent a way to do it. Let him perform several in his own way, and then suggest some method a little different from his, and nearer to the common method. If he readily comprehends it, he will be pleased with it, and adopt it. If he does not, his mind is not yet prepared for it, and should be allowed to continue his own way longer, and then it should be suggested again. 
Although most educators today haven’t heard about Colburn’s work, the impact of the IBL movement of his time was profound. The teaching of arithmetic was transformed across the nation (as was documented in the first issue of the American Journal of Education in 1826.) No longer did teenage boys struggle through copybooks to learn the “vulgar arithmetic”. Instead children would begin to study arithmetic at earlier ages (usually beginning around 7 in the latter half of the 19th century.) Although direct instruction followed by practice did become more common later, the notion that children are capable of and should understand reasons why the rules of arithmetic are what they are became a fixture of educational practice 150 years ago. The fact that all younger children, including girls, would subsequently study arithmetic in the US (and not an elite few as was common in the previous century) is due in large part to the success of the IBL movement initiated in Colburn’s time. Mathematics became a subject for everyone to study! This may very well be one of the most dramatic events in the history of mathematics education, having enormous social implications for the nation. Warren Colburn attributed his ideas about the role of inquiry in mathematics education to the Swiss educator Johann Pestalozzi (17261827) and the theme of engaging students actively in inquiry has recurred often in the history of education. Pestalozzi set up schools where IBL was practiced across disciplines over 200 years ago. In the early 20th Century, educators such as Maria Montessori (18701952) and John Dewey (18591952) advocated far greater autonomy for all their students to allow them to direct their own studies and construct their own understandings. Their focus was not solely mathematics, and the extent to which one might regard their proposals as IBL are debated by some, but the direction of motion is clear. They clearly believed inquiry was critical to a child’s mathematical development. The notion that students construct their own understandings by actively making sense of problematic situations is part and parcel of their programs. The success in particular of the Montessori School system across the world and in the US is well recognized. The Swiss psychologist Jean Piaget (18961980) probably stands out as the giant among 20th century educators for his research on the development of understanding in both children and adults. His quote “to understand is to invent” (which is also part of a title of one of his many books) may be the most widely used statement used by advocates of IBL approaches in education across disciplines. He studied children’s acquisition of number concepts, when they could successfully engage in deductive reasoning, the robust nature of misconceptions in math and science, and how the learner’s construction of ideas is needed to overcome such misconceptions. He demonstrated developmental stages are a necessary part of the learning process. Piaget was also well educated in advanced mathematics and was interested in the foundational work of the famous pure mathematician Grothendieck. His work has been refined and its general thrust validated by educators and psychologists over the past 50 years. While it isn’t possible to summarize even a small fraction of his work here, we note that his research (and that of many others building upon his work) provides the experimental justification for many of the educational practices that generally fit under the IBL umbrella. At the university level, R.L Moore is the first innovator we are aware of to systematize and use IBL in collegiate mathematics education. Beginning in the 1920’s and continuing for half a century at the University of Texas, he taught calculus via pure inquiry, not allowing his students to consult textbooks and in his classes had students construct their own derivations and proofs of results according to a sequence he devised. His work provided the inspiration for the EAF to fund all five IBL centers and his work has withstood the test of time, spanning the more than half a century. Moore’s work is particularly significant when one considers the prevalent view that because so much material has to be mastered in a short time at the collegiate level, one has no choice but to use a lecture format. He not only succeeded in dramatic ways in the calculus, but further, he demonstrated that graduate education could benefit as well. Moore directed 52 Ph. D. students, many of whom, and their subsequent students have continued his IBL tradition. Recent IBL Efforts. In 1989, the National Research Council issued a report, Everybody Counts: A Report to the Nation on the Future of Mathematics Education^{2},. The document discussed a national need for change and called for direct involvement of college mathematics educators. It says, “No reform of mathematics education is possible unless it begins with revitalization of undergraduate mathematics in both curriculum and teaching style.” (p. 39) The document notes that in the prior decades mathematics enrollments doubled while faculty grew by less than 30% and laments "Mathematicians seldom teach what they think about—and rarely think deeply about what they teach". It stresses the importance of Calculus as a gateway to the sciences and points out that as a large part of K12 mathematics is Calculus preparatory, changes in Calculus teaching will be needed if we expect K12 to change as well. About the process of teaching and learning Everybody Counts says “Evidence from many sources shows that the least effective mode for mathematics learning is the one that prevails in most of America’s classrooms: lecturing and listening” (p. 57) and adds a bit later 
In reality no one can teach mathematics. Effective teachers are those who can stimulate students to learn mathematics. Educational research offers compelling evidence that students learn mathematics well only when they construct their own mathematical understanding. (p. 58). 
But the document does not call for specific pedagogical changes other than to call for greater student engagement. For example, its Second Transition Principle says, “The teaching of mathematics is shifting form an authoritarian model based on ’transmission’ to a studentcentered practice featuring ‘stimulation of learning’.” The document does offer advice on how to bring about change, arguing against top down changes in favor of building up from the local level and calling for a national base of support for the effort. In K12, both within and outside the US educators have developed a variety curricula and approaches where the goal is for students to construct their own understandings through varying forms inquiry (ranging from very free investigations to more heavily structured problemsolving.) In 1985, California adopted a new Mathematics Framework which advocated teaching for understanding based on problem solving where is stated, “In problem solving the teacher should serve as a group facilitator rather than a direct leader.” In 1989 the National Council of Teachers of Mathematics produced its Curriculum and Evaluation Standards for School Mathematics, which also stressed that problem solving should play an integral part of instruction. To be sure, the mere fact that these documents were written did not translate into significant changes in practice in the US. However, it is clear that these documents recognized that the development of mathematical understanding requires substantial action on the part of the student to construct his/her ideas that direct instruction alone cannot provide. This vision of mathematics teaching and learning grew as researchers in mathematics education adopted more qualitative paradigms (for example, interview and case study methods) for their research, as opposed to experimental research where one compares pre and posttest measures to treatment and control groups. A nice summary of the views surrounding the NCTM movement by the mid 1990’s and its relation to mathematics education research is the paper Problem Solving as a Basis for Reform in Curriculum and Instruction: The Case of Mathematics, by James Hiebert, Thomas Carpenter, et al (Educational Researcher 25 (4) 1221 1996). They view mathematics as an act of sensemaking which results from students dealing with problematic situations. They trace the roots of their vision to Dewey and Brownell and relate it to research in mathematics education during the previous decades. Also in the mid 1990’s, the video analyses of US, German and Japanese 8^{th} grade classrooms as part of TIMSS were released (coordinated by Jim Stigler of UCLA) which showed quite dramatically, that even though US teachers said they were aware of the NCTM calls for change, instruction in the US had not changed substantially and the dominant instructional practice in K12 mathematics remained direct instruction followed by drill and practice. This is still true today. During the 1990’s the National Science Foundation became deeply involved in mathematics and science education. The NSF sponsored curriculum development projects K16 as well as numerous “systemic change projects” which were to ensure that the “reform” takes root. Elementary school materials, middle school materials, high school materials, and calculus materials were developed using NSF funding and then published and marketed across the US. Several hundred million dollars were spent on these programs, and many of the materials are available and in use today (and although it would be wrong to give the impression they have a large market share, they are significant players in some communities.) Most of the NSF materials distinguish themselves from their predecessors in that expect more inquiry and problem solving on the part of students, although to varying degrees. At the level of the Calculus one sees nothing that resembles the approaches of R.L Moore. In fact, some might even argue that the NSFfunded calculus projects represent the antithesis of what Moore believed, because of a perceived lack of rigor in these programs (in particular lack of epsilondelta reasoning), and instead their reliance on heuristic reasoning rooted in applications. But there is still similarity in the common recognition that the student must be the responsible party for constructing the basic ideas if understanding is to be attained. The projects were successful in that they engaged a substantial and enthusiastic group of mathematicians in thinking about educational reform at the collegiate level. In 2000 the NCTM issued a revised Principles and Standards for School Mathematics after an extensive writing and rewriting project over three years and taking into account input from many professional organizations. This document was carefully worded to avoid controversy about pedagogy, but it does call for learning with understanding and it takes the position that students “learn more and learn better when they can take control of their learning by defining their goals and monitoring their progress” (p. 21). Problem solving plays a prominent role in the document’s recommendations and it includes sample inquirybased tasks and discusses the development of students’ mathematical reasoning abilities and understanding of proof. It further contains expectations for student work across preK12 (according to the bands, PreK2, 35, 68, 912.) As the PSSM is our only national document outlining a vision for K12 mathematics education, and because it represents consensus of our nations professional organization for K12 mathematics teaching, it is important to note that the vision of IBL articulated by the centers is compatible with and offers an important tool to accomplish the NCTM’s goals. In the past few years, national agencies have begun the process of reviewing the mathematics education reform movements of the 1990’s trying to determine what succeeded, what did not succeed and why^{3}. The success stories center around projects that provides sustained professional development for educators and that focus on student thinking. It takes time for beliefs about teaching and learning to change and even longer for practice to change. They point out that institutional barriers to change can be substantial so it is critical to address these as well. Outside the US. Although differences in the culture of teaching makes it difficult to compare US approaches with those outside the US, there is important evidence of the success of IBL approaches outside the US. For example, mathematics education in Japan has received a great deal of attention in the US recently, largely because of their high scores of international assessments (TIMSS, for example). The TIMSS contained an extensive video analysis of randomly selected 8th grade mathematics lessons from the US, Germany and Japan, which has provided baseline data on teaching practices in these three countries (this was coordinated by Jim Stigler of UCLA). One finds practices incorporating the most inquiry on the part of students in Japan. According to Stigler, the typical classroom lesson in Japan unfolds according to a different script than do US lessons. A typical Japanese lesson begins by recalling a concept discussed earlier, followed by posing a problem for students to investigate. After students work on the problem, multiple solution methods are examined and some closure is attained through a teacherled discussion. In the typical US lesson the teacher demonstrates a procedure and students then practice what they have been shown, often completing 25 exercises in the same time Japanese students work on a single thoughtprovoking question. The analysis of the TIMSS lessons by mathematicians found that a striking difference in expectations of student work in class: in Japanese classrooms students spent more than 40% of class time engaged in tasks classified as ”invent or think”, while in Germany and the US the time was less than 5%^{4}. An excellent sourcebook of Japanese lessons is the book, The Open Ended Approach: A New Proposal for Mathematics Teaching by Jerry Becker and Shigeru Shimada (NCTM 1997). It contains a collection of problems assigned daily and a landscape of anticipated student responses. Study of this book clarifies the Japanese approach, where students engage in inquiry about openended problems and then teachers are expected to pull the big ideas out of the collective ideas generated by the class. An important case study is the work of Jo Boaler in her book, Experiencing School Mathematics, Open University Press (1997), where she analyzes student work and achievement over a threeyear period in two innercity high schools in London. One high school used standard curriculum and at the second school the curriculum was based on openended projects where students would work in collaborative groups and prepare reports on their work every few weeks. In the first school observations showed students were on task and completed their work, while in the second school the program was at times disorganized although students had a positive attitude towards their work. Achievement on standardized assessments in the second school was comparable to the first, better in some categories, and achievement on an openended assessment was significantly higher in the second school. Although we would not want to compare the second school to the vision of IBL articulated in the Center’s project, the implications of Boaler’s research is pretty clear. Student achievement is significantly enhanced by systematic opportunities to engage in inquiry. In the European Union, scores from The Netherlands are consistently among the highest. Dutch mathematics education has been guided in part by research conducted and curriculum developed at the Freudenthal Institute (named in honor off Hans Freudenthal 19051990 the leading Dutch mathematics educator of the 20th century). They call their work Realistic Mathematics Education and it uses contextbased problem solving with an emphasis on studentconstructed representation. (They also collaborated with the University of Wisconsin on their NSF funded project that led to the development of Math in Context, a grade 58 program published in the late 1990’s by Encyclopedia Britannica.) The progress in the Netherlands motivates much of the K12 work done by the UCSB center, in collaboration with the work of Cathy Fosnot and Maarten Dolk in Mathematics in the City. Three volumes by Fosnot and Dolk, Young Mathematicians at Work, outline their work and are available from Heinemann publishers. Concluding Remarks. The discussion here has considered a variety of mathematics education reform efforts. They inform the work of the Center’s project in several ways. First, here is considerable evidence form a variety of perspectives that students who engage in mathematical inquiry develop deeper and lasting understanding of the big ideas. This is true at all levels. Second, the dominant educational approach in the US mathematics is lecture followed by practice and these approaches are deeply embedded in educational culture and therefore are difficult to change. However, our conception of IBL combines the of inquiry pedagogy with the mathematical rigor our students need, and therefore it provides an important vehicle for transforming mathematics education at all levels. 
^{1}From Patricia Cohen, A Calculating People, The spread of Numeracy in Early America, Routledge (1999) p . 134 ^{2}Everybody Counts is online at http://www.nap.edu/books/0309039770/html. ^{3} Two such documents are Investigating the Influence of the Standards, National Research Council, National Academy Press (2002) and Foundations, Professional development That Supports School Mathematics Reform, National Science Foundation (2002) ^{4} For more details see J. Stigler, J Hiebert, Understanding and Improving Classroom Mathematics Instruction: An Overview of the TIMSS Video Study, Phi Delta Kappan, September 1997 1421. 