What are the best ways to teach math content, not just procedures, in grades K-6?

Thank you for the question you submitted to our REL Reference Desk regarding teaching math content in grades K-6. We have prepared the following memo with research references to help answer your question. For each reference, we provide an abstract, excerpt, or summary written by the study's author or publisher. The references are selected from the most commonly used research resources and may not be comprehensive. References are listed in alphabetical order, not necessarily in order of relevance. Other relevant studies may exist. We have not evaluated the quality of these references, but provide them for your information only.

Research References

1. Ballantyne, A. (2019). Untangling the math debate. BU Journal of Graduate Studies in Education, 11(2), 52-56.
   Retrieved from: https://eric.ed.gov/?id=EJ1230199
   From the abstract: “The question of how best to teach mathematics has been up for debate for decades. Traditionalists push for a back-to-basics type education, while reformers seek to teach students for understanding. At the same time, many teachers are dealing with their own feelings of anxiety about math. While it often appears that this debate must end in an either-or solution, perhaps the best way forward would be to seek a balanced solution. Teachers could be supported through this process of change with a combination of quality professional development and opportunities to engage in professional learning communities. Finding a way to strike a balance and end this debate will give teachers the opportunity to provide their students with a comprehensive mathematics education.”
2. Blazar, D. (2015). Effective teaching in elementary mathematics: Identifying classroom practices that support student achievement. Economics of Education Review, 48, 16-29. Retrieved from: https://www.sciencedirect.com/science/article/abs/pii/S0272775715000710
   Full text available at https://cepr.harvard.edu/files/cepr/files/blazar_2015_effective_teaching_in_elementary_m athematics_eer.pdf
   From the abstract: “Recent investigations into the education production function have moved beyond traditional teacher inputs, such as education, certification, and salary, focusing instead on observational measures of teaching practice. However, challenges to identification mean that this work has yet to coalesce around specific instructional dimensions that increase student achievement. I build on this discussion by exploiting within-school, between-grade, and cross-cohort variation in scores from two observation instruments; further, I condition on a uniquely rich set of teacher characteristics, practices, and skills. Findings indicate that inquiry-oriented instruction positively predicts student achievement. Content errors and imprecisions are negatively related, though these estimates are sensitive to the set of covariates included in the model. Two other dimensions of instruction, classroom emotional support and classroom organization, are not related to this outcome. Findings can inform recruitment and development efforts aimed at improving the quality of the teacher workforce.”
3. Doabler, C. T., Cary, M. S., Jungjohann, K., Clarke, B., Fien, H., Baker, S., ... & Chard, D. (2012). Enhancing core mathematics instruction for students at risk for mathematics disabilities. Teaching Exceptional Children, 44(4), 48-57.
   Retrieved from: https://eric.ed.gov/?id=EJ977326
   Full text available at https://journals.sagepub.com/doi/pdf/10.1177/004005991204400405?casa_token=0fpDm
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   From the abstract: “This paper presents eight practical guidelines that teachers can use to make core instruction more systematic and explicit for students with or at-risk for mathematics disabilities. In the paper, we use the notion of explicit and systematic instruction as a foundation for intensifying core math instruction. Explicit and systematic core instruction includes sufficient coverage of the most critical content areas of math and reflects current research on effective math instruction. To illustrate these points, we describe the core kindergarten curriculum used in our research, the Early Learning in Mathematics program. We conclude by modifying a lesson drawn from a popular core math program to demonstrate how teachers can use the guidelines with their existing curriculum. This information is particularly relevant to special educators because they can support schools in enhancing the quality of core math instruction.”
4. Doabler, C. T., & Fien, H. (2013). Explicit mathematics instruction: What teachers can do for teaching students with mathematics difficulties. Intervention in School and Clinic, 48(5), 276-285. Retrieved from: https://eric.ed.gov/?id=EJ1011763
   Full text available at https://journals.sagepub.com/doi/pdf/10.1177/1053451212473151?casa_token=ngZcHpK
   HPqAAAAAA:8XcUk4t9EqCzCBvDFsfi4hmXmsP8SvgFp1OqBLpOf0a5U7199G3BSW Mf5AfS6T5fSFlys8ZPg6-7JQ
   From the abstract: “This article describes the essential instructional elements necessary for delivering explicit mathematics instruction to students with mathematics difficulties. Mathematics intervention research indicates that explicit instruction is one of the most effective instructional approaches for teaching students with or at risk for math difficulties. Explicit instruction is a systematic approach that facilitates important instructional interactions between teachers and students around critical math content. This article describes a framework for delivering explicit math instruction in the early grades. Although the explicit framework is relevant across the range of early math content (i.e., measurement, geometry), the focus is on explicit instruction in the context of teaching place value concepts in kindergarten and first grade classrooms. Place value is a critical component of whole-number understanding and a necessary concept for students to develop mathematical proficiency. Key questions associated with explicit math instruction are also addressed.”
5. Doabler, C. T., Fien, H., Nelson-Walker, N. J., & Baker, S. K. (2012). Evaluating three elementary mathematics programs for presence of eight research-based instructional design principles. Learning Disability Quarterly, 35(4), 200-211. Retrieved from: https://eric.ed.gov/?id=EJ980671
   Full text available at https://journals.sagepub.com/doi/pdf/10.1177/0731948712438557?casa_token=TJBcCBo FIKUAAAAA:
   RwtuO0HtAAHQI1xXmFi6Dsd055VlI4I_ny0_1diFc9bN0w2sJiPcMMLk qeXaneGOgb2EvKrSkUKbgg
   From the abstract: “The present review builds on earlier research that evaluated the curricular features of core math programs to improve the performances of students with or at risk for mathematics difficulties. In this review, three elementary math programs, at Grades 2 and 4, were evaluated for the presence of eight instructional principles. Math intervention studies have empirically validated these principles for promoting math proficiency of students struggling with mathematics. Data were collected via a researcher-developed scoring rubric. Findings indicate that adherence to the instructional principles varied markedly within and across programs. In addition, the results indicated that the current textbooks contain a general lack of explicit instruction and provide too few practice opportunities to teach material to mastery. Implications for future curricular reviews and enhancing core math instruction are discussed.”
6. Frye, D., Baroody, A. J., Burchinal, M., Carver, S. M., Jordan, N. C., & McDowell, J. (2013). Teaching Math to Young Children. Educator's Practice Guide. What Works Clearinghouse. NCEE 2014-4005. What Works Clearinghouse. Retrieved from: https://eric.ed.gov/?id=ED544376
   From the abstract: “The goal of this practice guide is to offer educators specific, evidence-based recommendations that address the challenge of teaching early math to children ages 3 to 6. The guide provides practical, clear information on critical topics related to teaching early math and is based on the best available evidence as judged by the authors. The guide is organized around five recommendations: (1) Teach number and operations using a developmental progression; (2) Teach geometry, patterns, measurement, and data analysis using a developmental progression; (3) Use progress monitoring to ensure that math instruction builds on what each child knows; (4) Teach children to view and describe their world mathematically; and (5) Dedicate time each day to teaching math, and integrate math instruction throughout the school day.”
7. Fuchs, L. S., Bucka, N., Clarke, B., Dougherty, B., Jordan, N. C., Karp, K. S., ... & Morgan, S. (2021). Assisting students struggling with mathematics: Intervention in the elementary grades. Educator's practice guide. WWC 2021006. What Works Clearinghouse. Retrieved from: https://eric.ed.gov/?id=ED611018
   From the abstract: “Recent intervention research has demonstrated success in raising the achievement level of students who are struggling with mathematics. This practice guide, developed by the What Works Clearinghouse™ (WWC) in conjunction with an expert panel, distills this contemporary mathematics intervention research into six easily comprehensible and practical recommendations for teachers to use when teaching elementary students in intervention settings. Two federal laws, the Elementary and Secondary Education Act (ESEA) and Individuals with Disabilities Education Act (IDEA), require use of instructional practices supported by evidence. The recommendations presented in the guide address these laws by translating the body of high-quality evidence into actionable practices for teachers to use with their students. Although this guide is an update of the 2009 guide, "Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle School," it is narrower in scope, focusing only on practices and principles underlying effective small-group interventions in grades K-6. The practices that appear in this guide's recommendations highlight effective approaches to mathematics intervention that meet the needs of the students in small-group or one-on-one settings. Each of these practices move students toward more fluent performance of mathematics. Each recommendation includes features of intervention and/or instructional practices, with guidance on how to implement them, advice on how to overcome potential obstacles, and a short summary of the research evidence that supports the recommendation. This guide is designed to be used by teachers providing mathematics intervention to students who are struggling. This professional group includes special educators, mathematics general education teachers, mathematics specialists, and mathematics coaches.”
8. Powell, S. R., Stevens, E. A., & Hughes, E. M. (2019). Math language in middle school: Be more specific. TEACHING Exceptional Children, 51(4), 286-295. Retrieved from: https://eric.ed.gov/?id=EJ12087401
   Full text available at https://journals.sagepub.com/doi/full/10.1177/0040059918808762
   From the abstract: “Many educators use informal math language to make the content more accessible for middle school students, yet this use of informal language may have unintended consequences. Informal language may hinder students' development of a deep math lexicon and understanding of concepts and procedures across grade levels. Becoming proficient with math language is likened to learning a second language (Wakefield, 2000). In spite of this challenge, proficiency with the language of math is necessary to learning and communicating about math concepts and procedures (Schleppegrell, 2007). As such, educators must be mindful and purposeful about the language used to teach math. Through the authors' research experiences in schools and professional development opportunities with educators, they have observed two ways in which math language could be improved in middle school, which is the focus of this article. First, educators can use formal language instead of informal language. The authors provide examples for instances in which educators can use formalized math language in the "Instead of That, Say This" figures. Second, educators can be more precise with math terms that are closely related but have distinct meanings and characteristics. The authors provide examples of terms in which educators can use specific math language in the "Be Precise" figures. The authors focus on formalized math language and term specificity in two major areas of math: (a) numbers and operations and (b) geometry and measurement. They emphasize these two areas because numbers, operations, geometry, and measurement strands account for a majority of math standards in middle schools in the United States. At the end of the article, the authors provide suggestions for supporting students' learning of math language.”

Additional Organizations to Consult

• Association of Mathematics Teacher Educators (AMTE): https://www.amte.net/
  From the website: “The Association of Mathematics Teacher Educators (AMTE) is the largest professional organization devoted to the improvement of mathematics teacher education—it includes over 1,000 members devoted to the preservice education and professional development of K-12 teachers of mathematics.”
• National Council of Teachers of Mathematics (NCTM): https://www.nctm.org/
  From the website: “Founded in 1920, the National Council of Teachers of Mathematics (NCTM) is the world's largest mathematics education organization. The National Council of Teachers of Mathematics advocates for high-quality mathematics teaching and learning for each and every student.”

Methods:

Search Strings. Teaching math content grades K-6 OR ways to teach math content elementary middle school OR teaching math content curriculum effectiveness

Searched Databases and Resources.

• ERIC
• Academic Databases (e.g., EBSCO databases, JSTOR database, ProQuest, Google Scholar)
• Commercial search engines (e.g., Google)
• Institute of Education Sciences Resources

Reference Search and Selection Criteria. The following factors are considered when selecting references:

• Date of Publication: Priority is given to references published in the past 10 years.
• Search Priorities of Reference Sources: ERIC, other academic databases, Institute of Education Sciences Resources, and other resources including general internet searches.
• Methodology: Priority is given to the most rigorous study types, such as randomized controlled trials and quasi-experimental designs, as well as to correlational designs, descriptive analyses, mixed methods and literature reviews. Other considerations include the target population and sample, including their relevance to the question, generalizability, and general quality.