Inquiry-Based Instruction versus Example-Based Instruction The National Council of Teachers of Mathematics (NCTM) adopted new mathematics standards in 1989, changing the way math was traditionally taught in the United States. Below is a comparison of NCTM’s inquiry-based math instruction to high-quality example-based math instruction that has been proven to be effective for teaching all children mathematics.

Inquiry-based instruction Example-based instruction Consequences A “spiraling” method is used to bounce quickly from topic to topic, returning each year to the same topics. Students are expected to learn each topic eventually, after repeated short exposures.

The curriculum includes a logical progression of fewer topics that are taught to mastery and build on one another. These topics create a solid foundation for learning in subsequent years.

With the “spiraling” method, teachers are unconcerned about students mastering topics, figuring they will learn it the next year. The result is that many students fall between the cracks, never mastering basic math skills.

Students are required to work in groups to discover their own methods to solve problems, including unfamiliar ones. Teachers act as “facilitators” rather than instructors.

Teachers instruct students in using the most efficient algorithms and methods. Students usually work individually to solve practice problems.

Social issues make group work increasingly ineffective in middle and high school. Strong students are often placed in weak groups to help them, but weaker students often gossip or disengage. Strong students learn despite their frustration with the slow pace of group discovery, while weak students learn little.

Teachers encourage students to discover patterns, and must be highly skilled in mathematics to decide if student discovery is mathematically correct or not. Texts lack content.

Teachers demonstrate patterns and working examples. Texts are a resource for learning, and support even the weakest teacher with strong mathematical content.

School districts have great difficulty hiring teachers proficient in mathematics, especially at the elementary school level. The result is students leaving inquiry-based classrooms with incomplete or inaccurate solutions to problems.

Students use pictures, words, and/or physical models to show or explain the process.

Students and teachers write step-by-step procedures or mathematical proofs to show process.

Students using inquiry-based curricula are illiterate in the language of mathematics, and miss out on learning the logic involved in mathematically-accurate procedures and proofs.

Textbooks encourage “exploration,” so do not provide explanations or examples. No problems are supplied with answers in the back of the book for self-check of understanding.

Textbooks are a resource for extracting information and include explanations, proofs, and examples. Answers are offered for some problems at the back of the book for self-check of understanding.

Students using inquiry-based curricula lack the skill of using a textbook as a tool for learning.

The discovery process is very time-consuming, leaving little time for students to practice for proficiency before moving on to the next topic.

Students are instructed in math concepts and problem solving methods, then practice for proficiency.

Students using inquiry-based curricula do not become computationally proficient and have difficulty recalling procedures used in solving problems previously studied. This results in the need to rediscover algorithms, even for trivial problems.

“Real world” problems are very time-consuming and include many non-mathematical elements, like writing about the solution, drawing pictures, gluing and pasting, etc. Problems are too simplistic, and by sixth grade students are two years behind students in top-performing countries in math.

“Real world” problems are usually easily solved within 15 minutes, so several are assigned to develop proficiency in various types of problems based on the mathematical concept student is learning. They are based on a solid foundation of fundamental skills and computational fluency. Mathematical rigor increases over time.

Students in example-based instruction are superior problem solvers to those in inquiry-based instruction, due to the fact that they have a larger base of knowledge and well-practiced problem solving skills.

Math is more about the process than the product.

Math is about the product and includes process.

Inquiry-based instruction teaches that it is okay to get a wrong answer if the thought processes are sound. Example-based students are looking for accuracy, even to the point where they are expected to account for margin of error.

Proponents believe that it is harmful to students to perform algorithms before they have complete understanding, thus understanding must precede knowledge.

Proponents believe that learning algorithms and gaining conceptual understanding go hand-in-hand, thus knowledge precedes and fosters understanding.

Inquiry-based students are not taught the most efficient algorithms, thus they are at a serious disadvantage in college when those algorithms are expected to have been learned with mastery. Example-based students master algorithms, and, in time, gain understanding through usage to solve problems.

Calculators are used liberally. Arithmetic computation of numbers larger than 2-3 digit integers is not done. Using calculators, like watching television, is a passive activity, using little brain function.

Calculator use is limited. Proficiency in arithmetic computation is encouraged, to develop deep, long-term knowledge and understanding of arithmetic operations. Computing with paper and pencil, like reading a book, is active, resulting in increased brain function and skill building.

Inquiry-based students lack development of deep, long-term knowledge and understanding of arithmetic operations, due to lack of practice in computation and use of algorithms. Computational fluency is also required to learn algebra efficiently.

A balanced math curriculum has three pillars:

computational fluency (knowledge), conceptual understanding, and problem solving skills.

Inquiry-based instruction leaves out computational fluency and therefore loses its balance and its foundation. Without computational fluency, students cannot develop permanent (long term memory) conceptual understanding and do not develop good problem solving skills. Mathematical reasoning and understanding come from computational fluency. The, "what must I find to solve this problem?" comes from experience, not repeated isolated discovery marathons.