Math Facts

Math Facts Development

Standard topics in arithmetic— acquisition of basic math facts and fluency with whole-number operations— are covered extensively in Math Trailblazers. The goal of the math facts development in Math Trailblazers is for students to learn the basic facts efficiently, gain fluency with their use, and retain that fluency over time. A large body of research supports an approach in which students develop strategies for figuring out the facts rather than relying on rote memorization. This not only leads to more effective learning and better retention, but also to the development of mental math skills.

Generally, students move through three developmental stages when acquiring operational understanding and fluency with the math facts:

Direct modeling, in which students recreate the action in the problem using manipulatives;
Counting strategies, such as counting on and counting back; and
Reasoning from known facts, in which students work from facts they already know. If, for example, a student knows that 6 + 4 = 10, then he or she has a quick way to access 7 + 4 (Carpenter, 1999, National Research Council, 2001).

Three Developmental Stages When Acquiring Math Facts

Throughout the curriculum, students are encouraged to invent their own strategies and to share their thinking with one another. However, particular emphasis is given to specific reasoning strategies.

Math Facts Practice and Assessment

In developing the Math Trailblazers math facts program, a careful balance was created between strategies and drill. Practice and assessment of the math facts is based on the results of research in the field. The following paragraph from Adding It Up: Helping Children Learn Mathematics published by the National Research Council summarizes much of the literature on practice:

“Practicing single-digit calculations is essential for developing fluency with them. This practice can occur in many different contexts, including solving word problems. Drill alone does not develop mastery of single-digit combinations. Practice that follows substantial initial experiences that support understanding and emphasize ‘thinking strategies’ has been shown to improve student achievement with single-digit calculations”(National Research Council, 2001).

Researchers caution that too much drill too soon can have a negative impact on students’ acquisition of the math facts. Studies have shown that “premature practice can be detrimental but that properly managed practice is essential in the development of expertise. . . . Brief, engaging, and purposeful practice distributed over time is usually most effective” (Brownell & Chazal, 1935; Hiebert, 1999; Isaacs & Carroll, 1999).

Following these recommendations, fact practice in Math Trailblazers is characterized by the distributed practice of small groups of facts that lend themselves to specific strategies. Practice and assessment occur after students have had substantial experiences learning the underlying concepts and developing strategies that make sense to them.

The teaching of the basic facts in Math Trailblazers is characterized by the following elements:

Use of StrategiesUse of Strategies. Students first approach the basic facts as problems to be solved rather than as facts to be memorized. In all grades, students are encouraged to use strategies to find facts, so they become confident that they can find answers to facts problems that they do not immediately recall. In this way, students learn that math is more than memorizing facts and rules that “you either get or you don’t .”

Distributed Facts Practice. Students study small groups of facts that can be found using similar strategies .

Practicing the Multiplication Facts in Small Groups in Grade 3

Practice in Context. Students continue to practice all the facts as they use them to solve problems, investigate math concepts, and play math games.

Appropriate Assessment. Students are regularly assessed to see if they can find answers to facts problems quickly and accurately and retain this skill over time. They take a short quiz on each group of facts. Students record their progress on Facts I Know charts and determine which facts they need to study.

A Multiyear Approach. In Grades 1 and 2, the curriculum emphasizes the use of strategies that enable students to develop proficiency with addition and subtraction facts by the end of second grade. In Grade 3, students review the subtraction facts and develop proficiency with the multiplication facts. In Grade 4, the addition and subtraction facts are checked, the multiplication facts are reviewed, and students develop fluency with the division facts. In Grade 5, students review the multiplication and division facts.

Facts Will Not Act as Gatekeepers. Use of strategies and calculators allow students to continue to work on interesting problems and experiments while learning the facts. Lacking quick recall of the facts does not prevent students from learning more complex mathematics.


Review and Practice

Every unit includes opportunities for distributed practice of concepts and skills. These resources can be found primarily in two places:

• Daily Practice and Problems (DPP )

Systematic Practice and Assessment of Facts in Daily Practice and Problems
• Home Practice

Math Fact Practice in Home Practice

Daily Practice and Problems is a set of short exercises incorporated in the daily routine that provide ongoing review and study of math concepts and skills and provides a structure for systematically reviewing basic math facts.

Home Practice is a series of problems that supplement the homework included in the lessons. The Home Practice distributes skills practice throughout the units and reviews concepts studied in previous units.

Math Facts in Grade K
Math Facts in Grade 1
Math Facts in Grade 2
Math Facts in Grade 3
Math Facts in Grade 4
Math Facts in Grade 5



Learn more about the research that supports Math Trailblazer’s approach to the Math Facts and the shift away from timed fact tests:


  • Baroody, A.J. 2006 “Why Children Have Difficulties Mastering the Basic Number Combinations and How to Help Them.” Teaching Children Mathematics 13 (August): 22–31.
  • Boaler, Jo. 2015 “Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts.” YouCubed. Stanford University.
  • Boaler, Jo. 2014 “Research Suggests Timed Tests Cause Math Anxiety.” Teaching Children Mathematics. 20 (April): 469–474.
  • Kling, G. and Bay-Williams, J.M. 2014 “Assessing Basic Fact Fluency.” Teaching Children Mathematics. 20(April): 488–497.
  • National Research Council. “Developing Proficiency with Whole Numbers.” In Adding It Up: Helping Children Learn Mathematics. J. Kilpatrick, J. Swafford, and B. Findell, (Eds.). National Academy Press, Washington, DC, 2001.