I have recently heard several mathematicians claim that the educational philosophies of Math Circles and the Inquiry Learning Community are essentially the same. I disagree. I will contrast the differences between these two approaches, along with two other common educational philosophies in the United States. All four approaches to math education differ significantly both in terms of the overall instructional goals and in terms of the primary methods used to achieve these goals.

In my experience, even the originators and staunchest advocates for specific philosophies incorporate the other approaches when putting their favored one into practice. Instructors should think carefully about the goals they have for a given set of students, and then choose a combination of approaches they believe most likely to meet those goals.

I would like to invite you to comment on any thoughts that you have about these lists. A few questions that I have for readers include:

- Which blend of these pedagogical approaches have you found congenial for specific audiences?
- Are there other approaches that are essentially different that I did not include in this list?
- What steps do you find helpful when transitioning students who are used to one pedagogical approach to another?
- There is a tendency for students from high poverty schools to be exposed primarily to Traditional Math approaches. The Common Core State Standards represent an effort to slightly improve on this prevailing norm, by pushing teachers in the direction of Conceptual Math instruction. What do you think people who love math and teaching should do to improve access to high quality math education for these students?

I look forward to hearing your ideas!

### Traditional Math

The primary goal of the Traditional Math approach is to teach students to solve problems of a specified type as easily and efficiently as possible. This approach arose out of a need to broaden the pool of people able to accurately perform specific computations.

Because of these goals, Traditional Math instruction has the following characteristics:

- Traditional Math instruction is efficient — the very simplest methods for solving the most common problems are taught directly to the students, whose only task is to remember what they have been told. The arrangement of topics leads students in an orderly way through the topics so that they have precisely the skills they need at each later point in the curriculum.

- Traditional Math instruction is controlled — the instructor does not need to take student responses into consideration when planning the content and order of the course, though he or she may respond to students who need help by adjusting the pace of the course slightly. There is no chance that the instructor or the students will contemplate a question that cannot immediately be answered by recourse to the authority of the text or the algorithm.

- Traditional Math instruction is straight-forward — problems do not have more than one possible answer or interpretation, there is always one that is designated as correct. Students never work on problems that take weeks or months to resolve.

- Traditional Math instruction is egalitarian — every aspect of every lesson is expected to be within the grasp of every student. All students are expected to gain the same knowledge from each lesson.

- Students memorize facts and algorithms without developing conceptual understanding. In K-12 education, the development of student strategies is actively discouraged in favor of memorization by instituting timed tests and by banishing manipulative aids such as the use of fingers. Cute sayings and songs are used to memorize facts.
- Students are eased into difficult topics by careful scaffolding, often divided into specific types of problems. Conceptually messy or difficult ideas are papered over by giving students a definition or rule for dealing with those situations.
- Providing practice so that students readily recall facts and perform procedures is more important than developing the ability of students to solve realistic, practical problems or to think logically and critically.
- Students work word problems which represent unrealistic, overly simplified situations. The problem statements include no extraneous information, provide all needed information, and usually involve only one math concept. Students are actively discouraged from reading and interpreting the text of these problems. Students are often instead encouraged to look in the section of the book they are in to find an example problem which is just like the one they are asked to solve. Students are told to focus only on certain key words which are supposed to tell them which operation to apply without thinking. They are discouraged from drawing accurate pictures representing the problem (even young students are told to draw only circles or a bar or tally marks to show the quantities efficiently). Older students are praised for not needing to draw any diagrams or pictures to aid their thinking. Students are not expected to write substantively to explain their thinking. Because the original context is not actually that important, students are not expected to re-contextualize their answers to consider whether they make sense in the real world.
- Students are trained that being good at math means being quick and not needing to think. Needing to struggle is a sign of a lack of intelligence, practice, or attentiveness.

### Conceptual Math (Common Core State Standards)

The primary goal of the Conceptual Math approach is to guide students to a deep enough understanding of common math topics that they can devise multiple approaches to solve those kinds of problems, and make sense out of their answers. In our current economy, employees need to know when to apply common computational approaches more than they need to know how to fluently perform multi-digit computations. Most employees are asked to devise ways to solve a range of problems rather than simply following a procedure laid out by someone else.

On the other hand, the Conceptual Math approach still aims to be accessible to all students and to all teachers. This means that course and lesson designs must be simple enough that teachers with many students and busy schedules can implement them easily. As a consequence, the Conceptual Math approach puts more emphasis on deep understanding of traditional math topics rather than developing the ability of students to research and tackle realistic practical problems or to create and tackle their own mathematical questions.

Because of these goals, Conceptual Math instruction has the following characteristics:

- Conceptual Math instruction is less efficient — it takes longer to get to a fluent procedure for solving a given kind of problem. However, the arrangement of topics in a course is still carefully chosen so that students are guided to an understanding of the course topics in an orderly way.
- Conceptual Math instruction is somewhat risky — the instructor must respond to the ideas of the students. The instructor must help them to see whether their ideas make sense and how they can be expressed using formal mathematical notation. There is a chance that a student will bring up an approach that is unknown to the instructor. However, the domain of inquiry is usually restrictive enough that most questions and interpretations can be anticipated.
- Conceptual Math instruction is fairly straight-forward — students sometimes tackle problems that have more than one possible answer or interpretation, but the domain of possibilities is usually somewhat restricted. Concepts do often take days, weeks, or even years to be developed, but individual problems usually are resolved within one day.
- Conceptual Math instruction is egalitarian — every aspect of every lesson is expected to be within the grasp of every student. While students may use different methods to arrive at answers, they all cover the same content.
- When learning a new concept, students are first introduced to concrete models and situations that illustrate the operation and they use manipulatives and drawings to solve problems. They then devise and discuss their own strategies based on place value, properties of operations, or an understanding of other math concepts. Finally, students learn algorithms and are expected to be able to justify why standard approaches work verbally and in writing.
- Students are given word problems which represent unrealistically simplified situations, usually involving unrealistic numbers and measurements, and usually expressed with a minimal number of sentences using simplistic vocabulary. The problem statements often do include extraneous information (often these problems are featured in a special section in the curriculum). Students are encouraged to read and interpret the text of the problems and are trained not to rely on keywords too much by the practice of inserting unknowns in various parts of the story rather than always at the end. However, students are usually trained to use generic circles, a bar, or tables to show quantities rather than freely representing them in whatever way makes sense to them. Students are expected to write substantively to explain their thinking. Students are also expected to re-contextualize their answers enough to decide whether they make sense.
- Instructors strive to respond to student questions by helping the students to explore them. This can be very challenging. When attempting to support the development of student strategies, instructors are often tempted to tell the students how they should be thinking and how they should record their thinking rather than allowing these processes to originate with the students. It is difficult to avoid teaching all of the strategies as if they were algorithms.
- Students are trained that being good at math means being willing and able to think carefully and explain their reasoning. Students are trained to believe that struggling is a sign that their brains are growing, but they do not usually struggle with any given concept for very long before they are rescued by a hint from an instructor. They are carefully supported through the process of learning by being given problems that are just within their zone of proximal development.

### Inquiry or Project Based Learning

The primary goal of the Inquiry approach is to teach students to create and investigate their own questions. This approach to instruction originated with those interested in preparing students to be scientists, engineers, programmers, or entrepreneurs.

The instructor often guides student inquiry by posing the initial question, which usually does not provide all of the needed information, and is deliberately badly defined. Problems often involve messy, realistic numbers. Students pose sub-questions and have substantial control over the direction their investigation will go. Students not only re-contextualize their results, but often present their results to outside audiences in a variety of written and verbal formats (including videos and web pages). During concluding discussions, the group creates anchor charts to codify strategies and facts they have discovered.

Communication and collaboration are explicit goals of the Inquiry approach. Students share their thinking verbally and in writing and give one another meaningful feedback. There is significant emphasis on teaching students about ways they can contribute positively to a team effort.

Because of these goals, the Inquiry approach to instruction has the following characteristics:

- Inquiry instruction is rather inefficient — lots of time is necessarily used to brainstorm, discuss, decide, and resolve conflicts. Excellent classroom management and organizational skills on the part of the instructor are necessary to avoid complete chaos or students spending large amounts of time doing trivial tasks that support no real learning.
- Inquiry instruction is very risky — it is likely that students will come up with questions the instructor does not know how to address, or that they will pose questions that don’t make any sense, or that the class will fail to cover the objectives of the course by side-stepping the approach the instructor had in mind. Students have authority to invent definitions and procedures, but they may end up producing results which are mathematically incorrect or which lack the appropriate level of rigor. Coming up with good topics for a given group of students and ways to provide structure for students as they work is very challenging.
- Inquiry instruction is complicated — everything depends on the individual choices of the students and it can be difficult to manage a classroom full of divergent thinkers and personalities. Materials can be expensive and material management can be challenging. Problems and questions take a long time to resolve and careful planning and management is needed to avoid getting bogged down. Students can easily get stuck and give up in frustration.
- Inquiry instruction is not egalitarian — different students will learn different things as a result of their choices. Social and academic hierarchies will be more obvious in the classroom due to frequent group work.
- Students are provided with much less scaffolding than is typical in either Traditional Math or Conceptual Math approaches. Students must learn to devise methods of making complicated problems tractable. Students wrestle with conceptually messy and difficult ideas through small and large group discussions and experiments of their own devising.
- Students are trained that being good at math means being willing to stick with a complicated problem, struggling to find a solution, and being able to communicate their thinking to others. \item The process of developing, refining, investigating, solving, and presenting questions and their solutions is more important than the specific math topics studied. The history of mathematics as a discipline and cultural aspects of doing mathematics are not emphasized.

### Math Circles

The primary goal of the Math Circle approach to instruction is to teach students learn how to work creatively in the discipline of mathematics. They create new mathematical playgrounds, brainstorm new questions for existing mathematical playgrounds, make original approaches to questions posed, generate data for given approaches, design ways to organize information obtained, propose conjectures about patterns they see, seek proofs of conjectures, find ways to define terms that make it easier to explain results, and express their results using diagrams, mathematical notation, and terms the way a mathematician would. Students learn to seek connections between seemingly different situations.

One of the goals of a Math Circle is to enculturate students as mathematicians. Students cannot develop this culture on their own working in small groups, so a Math Circle instructor frequently models the norms of mathematical discourse. Most of the ideas for solving problems come from the students (though the instructor may ask leading questions when needed). However, the instructor frequently intrudes while students are presenting their ideas to impose the cultural norms of math as a discipline.

Students learn about mathematics as a discipline. They learn to value (and collect) failed attempts as an aid to eventually solving a problem. They practice common proof techniques, and learn to use terms and notation so that other mathematicians will understand what they say and write. Students are exposed to the history of the mathematical ideas they encounter. They also learn what makes a question mathematically interesting, and how to deal with being stuck (emotionally and mathematically). Students learn to interact appropriately with fellow researchers, including being able to communicate effectively in verbal and written form, balancing personal emotional needs against those of a group, building a collegial atmosphere capable of producing interesting mathematical insights, and enjoying the process of mathematical discovery.

Because of these goals, the Math Circle approach to instruction has the following characteristics:

- Math Circle instruction is deliberately inefficient — this approach is usually not concerned with mastery of specific math topics. However, when using this approach to teach externally specified content, a Math Circle instructor will seek a mathematically interesting question that will likely lead students past the content to be addressed. The question will not usually be directly about the content to be taught (although sometimes it is). It often takes weeks or months for students to resolve the many questions they pose along the way.
- Math Circle instruction is a high wire act — the instructor never knows where the students will go next. The instructor usually must do quite a bit of research into possible directions the discussion might go to scope out the landscape for possible scenic overlooks and pitfalls. However, the students might take off in another direction which the instructor has not prepared. No matter how much mathematics the instructor knows it is impossible to prepare for everything that might happen.
- Math Circle instruction is complicated — it is very challenging to orchestrate an intellectually satisfying conversation that allows all students to participate. The instructor must think hard to choose an opening gambit for the topic which is low entry, high ceiling, and compelling to the specific students in the class. It can be difficult to find good questions for a given group of students, especially if there are also specific content goals to incorporate.
- Math Circle instruction is not egalitarian — different students will learn different things. This is because the instructor is usually not careful to make sure that all students have mastered any of the content discussed. Some students will have mastered all aspects of the proofs discussed, and others may only have practiced some basic math skills while having fun looking for patterns. A Math Circle instructor chooses to keep aiming high even though not everyone may be ready to appreciate the nuances yet.
- Math Circle instructors tend not to choose topics that are straightforward applications of standard math concepts. This means that real-world problems that are uninteresting mathematically are not usually included. Real-world applications tend to come in only when the mathematics that emerges from those applications is intrinsically compelling.
- Math Circles drop students into the deep end — often giving them impossible or unsolved problems, often exposing them to concepts well beyond those typical for their grade level. This has nothing to do with the level of preparation of the students. The expectation of critical thinking and problem solving is simply set much higher than usual, and the instructor establishes the expectation that the group will need to struggle and might sometimes fail to answer some questions.
- Students are trained not to worry about being good at math. Mathematics is a huge discipline and it is impossible for anyone to know most of what there is to know. There will always be someone who knows more than you and who can work faster, and who cares anyway? We do mathematics because it is fun, beautiful, and is a means of solving interesting and important problems. We learn to appreciate the beauty of mathematical ideas and the joy of feeling our minds at work

I think that the teachers I learned the most from used mainly traditional or conceptional math teaching with a little bit of the others mixed in. I think that for us students this really helped keep us interested and learning in more then one way.

Wow, this is great, thank you! What a wonderful break-down of pedagogies. I especially like your last paragraph; “Students are trained not to worry about being good at math. Mathematics is a huge discipline and it is impossible for anyone to know most of what there is to know. There will always be someone who knows more than you and who can work faster, and who cares anyway? We do mathematics because it is fun, beautiful, and is a means of solving interesting and important problems. We learn to appreciate the beauty of mathematical ideas and the joy of feeling our minds at work.”

At first glance, this attitude seems universal, but then I realize that it depends on the audience. A highly motivated student who puts a ton of pressure on himself or herself to be perfect at the expense of creativity and perseverance would benefit from this; a student failing to put in the work required to be more efficient at basic calculations might take this as an excuse.