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Math

December 2020

Question

What are the benefits of mathematical instruction that teaches students to approach and solve problems using multiple methods? How does this educational approach benefit student learning as opposed to exclusively teaching the standard algorithm?

Response

Following an established research protocol, REL Central conducted a search for research reports as well as descriptive study articles to help answer the question. The resources included ERIC and other federally funded databases and organizations, research institutions, academic databases, and general Internet search engines. (For details, please see the methods section at the end of this memo.)

References are listed in alphabetical order, not necessarily in order of relevance. We have not evaluated the quality of the references provided in this response, and we offer them only for your information. We compiled the references from the most commonly used resources of research, but they are not comprehensive and other relevant sources may exist.

Research References

Clements, D. H., Agodini, R., & Harris, B. (2013). Instructional practices and student math achievement: Correlations from a study of math curricula (NCEE 2013-4020). U.S. Department of Education, Institute of Education Sciences, National Center for Education Evaluation and Regional Assistance. Retrieved from https://eric.ed.gov/?id=ED544189

From the ERIC abstract:

“This brief is directed to researchers and adds to the research base about instructional practices that are related to student achievement. Additional evidence on these relationships can suggest specific hypotheses for the future study of such instructional practices, which, in turn, will provide research evidence that could inform professional development of teachers and the writing of instructional materials. The results of this study revealed a pattern of relationships largely consistent with earlier research, but not in every case. Results that are consistent with previous research include increased student achievement associated with teachers dedicating more time to whole-class instruction, suggesting specific practices in response to students’ work (1st grade only), using more representations of mathematical ideas, asking the class if it agrees with a student’s answer, directing students to help one another understand mathematics, and differentiating curriculum for students above grade level (2nd grade only). Less consistent results were found in three 2nd-grade results, and include lower achievement associated with teachers’ higher frequency of eliciting multiple strategies and solutions; prompting a student to lead the class in a routine; and with students more frequently asking each other questions. These findings suggest that practices associated with higher achievement gains include both student-centered and teacher-directed practices; however, some student-centered practices were associated with lower achievement gains.”


Daro, P., Mosher, F. A., & Corcoran, T. (with Barrett, J., Battista, M., Clements, D., Confrey, J., Daro, V., Maloney, A., Nagakura, W., Petit, M., & Sarama, J.). (2011). Learning trajectories in mathematics: A foundation for standards, curriculum, assessment, and instruction (CPRE Research Report # RR-68). Consortium for Policy Research in Education. Retrieved from https://eric.ed.gov/?id=ED519792

From the ERIC abstract:

“The concept of learning progressions offers one promising approach to developing the knowledge needed to define the ‘track’ that students may be on, or should be on Learning progressions can inform teachers about what to expect from their students. They provide an empirical basis for choices about when to teach what to whom Learning progressions identify key waypoints along the path in which students’ knowledge and skills are likely to grow and develop in school subjects. Such waypoints could form the backbone for curriculum and instructionally meaningful assessments and performance standards. In mathematics education, these progressions are more commonly labeled learning trajectories. These trajectories are empirically supported hypotheses about the levels or waypoints of thinking, knowledge, and skill in using knowledge, that students are likely to go through as they learn mathematics and, one hopes, reach or exceed the common goals set for their learning. Trajectories involve hypotheses both about the order and nature of the steps in the growth of students’ mathematical understanding, and about the nature of the instructional experiences that might support them in moving step by step toward the goals of school mathematics. This report aims to provide a useful introduction to current work and thinking about learning trajectories for mathematics education; why everyone should care about these questions; and how to think about what is being attempted, casting some light on the varying, and perhaps confusing, ways in which the terms trajectory, progression, learning, teaching, and so on, are being used by the authors and their colleagues in this work.”


Gilbertson, N. J. (2019). Maintaining the mathematical focus of whole-class discussions: Dilemmas and instructional decisions. In S. Otten, A. G. Candela, Z. de Araujo, C. Haines, & C. Munter (Eds.), Proceedings of the forty-first annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1639–1648). University of Missouri. Retrieved from https://eric.ed.gov/?id=ED606791

From the abstract:

“As teachers shift their practice from traditional student-to-teacher interaction patterns to collaborative discussions around rich mathematics, they often encounter instructional challenges. One such challenge is deciding when to pursue interesting and productive ideas that run contrary to the particular mathematical goal of the lesson. In this research summary, I report results from a study involving one experienced teacher’s instructional decisions and how she maintained attention to the lesson-specific content goal given possible alternative pathways.”


King, S., & Campbell, T. (2019). Using interpersonal discourse in small group development of mathematical arguments. In S. Otten, A. G. Candela, Z. de Araujo, C. Haines, & C. Munter (Eds.), Proceedings of the forty-first annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1631–1638). University of Missouri. Retrieved from https://eric.ed.gov/?id=ED606932

From the abstract:

“The development of proofs and argumentation is one of the major standards for mathematical practices in K–12 education that researchers and practitioners alike are continuing to improve. Further, the use of discourse is considered essential in the learning of mathematical concepts at these levels. However, K–12 educators continue to confound how best to utilize student interpersonal discourse to advance the development of mathematical arguments. This qualitative study examines the nature of student discourse in small-group interactions as students create collective arguments based on mathematical evidence. In examining the patterns of discourse in small groups, the study concludes the effectiveness of various types of discourse in peer-to-peer interaction as students develop more analytical thoughts through the support of their discourse with one another in creating proofs and arguments.”


Sharp, L. A., Bonjour, G. L. & Cox, Ernest. (2019). Implementing the math workshop approach: An examination of perspectives among elementary, middle, and high school teachers. International Journal of Instruction, 12(1), 69–82. Retrieved from https://eric.ed.gov/?id=EJ1201353

From the abstract:

“Improving student performance in mathematics is reinforced through the use of effective teaching practices. One such practice, the math workshop approach, is a rigorous, student-centered way to teach mathematics that fosters inquiry among a community of learners. The purpose of the present study was to explore the perspectives of mathematics teachers who implemented the math workshop approach in their classrooms. Using a concurrent mixed methods research design, an electronic questionnaire was administered among four elementary teachers, two middle school teachers, and two high school teachers who had several years of teaching experiences. Quantitative and qualitative data were collected and analyzed separately to examine congruence with reported perspectives. Quantitative data were tabulated and reported with frequencies and percentages. Qualitative data were analyzed with descriptive analysis techniques to identify themes. Findings revealed that participants recognized the math workshop approach as an effective teaching practice to improve teaching and learning in mathematics. These findings pointed to implications for teacher preparation programs and professional learning efforts among math professionals employed within school districts. Limitations and future areas of study were also discussed.”


Schumacher, R., Taylor, M. J., & Dougherty, B. (with Woodward, J.). (2019). Professional learning community: Improving mathematical problem solving for students in grades 4 through 8; Facilitator’s guide (REL 2019-002). U.S. Department of Education, Institute of Education Sciences, National Center for Education Evaluation and Regional Assistance, Regional Educational Laboratory Southeast. Retrieved from https://eric.ed.gov/?id=ED595201

From the ERIC abstract:

“REL Southeast developed this facilitator’s guide on the topic of mathematical problem solving for use in professional learning community (PLC) settings. The facilitator’s guide is a set of professional development materials designed to supplement the What Works Clearinghouse practice guide, Improving Mathematical Problem Solving in Grades 4 Through 8 (Woodward et al., 2012). The practice guide provides research-based recommendations for teachers to incorporate into their classroom practice. The facilitator’s guide is designed to complement and extend the practice guide by providing teachers in a PLC setting with additional, step-by-step guidance for the best ways to implement some of these evidence-based recommendations. The facilitator’s guide focuses on three of the five recommendations from the mathematics problem solving practice guide to ensure in-depth coverage of the topics and to provide ample practice opportunities and time for reflection. The three practice guide recommendations on which the facilitator’s guide is based are: teach students how to use visual representations (Recommendation 3), expose students to multiple problem-solving strategies (Recommendation 4), and help students recognize and articulate mathematical concepts and notation (Recommendation 5). REL Southeast chose these three recommendations because they are interrelated and include critical content to address the two high-leverage regional needs communicated by the Improving Mathematics Instruction Research Alliance which include improving classroom discourse in mathematics and enhancing students’ mathematical problem-solving skills.”


Star, J. R., Caronongan, P., Foegen, A., Furgeson, J., Keating, B., Larson, M. R., Lyskawa, J., McCallum, W. G., Porath, J., & Zbiek, R. M. (2015). Teaching strategies for improving algebra knowledge in middle and high school students (NCEE 2015-4010). U.S. Department of Education, Institute of Education Sciences, National Center for Education Evaluation and Regional Assistance. Retrieved from https://eric.ed.gov/?id=ED555576

From the ERIC abstract:

“Mastering algebra is important for future math and postsecondary success. Educators will find practical recommendations for how to improve algebra instruction in the What Works Clearinghouse (WWC) practice guide, ‘Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students.’ The methods and examples included in the guide focus on helping students analyze solved problems, recognize structure, and utilize alternative approaches to solving algebra problems. Each recommendation includes the level of supporting research evidence behind it, examples to use in class, and solutions to potential implementation roadblocks. Teachers can implement these strategies in conjunction with existing standards or curricula. In addition, these strategies can be utilized for all students learning algebra in grades 6–12 and in diverse contexts, including during both formative and summative assessment. Administrators and professional development providers can use the guide to implement evidence-based instruction and align instruction with state standards or to prompt teacher discussion in professional learning communities.”



Additional Resources to Consult

Common Core State Standards Initiative, Standards for Mathematical Practice: http://www.corestandards.org/Math/Practice/

From the website:

“The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important ‘processes and proficiencies’ with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).”

Heinemann, Contexts for Learning Mathematics: https://www.heinemann.com/contextsforlearning/

From the website:

“Math fact drills and endless word problem worksheets are ineffective and meaningless to students. Contexts for Learning Mathematics provides carefully crafted units designed to foster deep understanding in a math workshop environment. By setting each unit in the context of students’ lives, teaching and learning immediately becomes engaging and meaningful in your classroom.”

What Works Clearinghouse, What Works in Math: https://ies.ed.gov/ncee/wwc/Math/

From the website:

“There’s no single answer to that broad question. Instead, what works varies by grade, subject, and even delivery model. WWC products allow educators to better understand what works in different contexts. WWC intervention reports show which tools increase mathematics achievement by grade, while WWC practice guides show effective practices for topics such as fractions.”



Methods

Keywords and Strings

The following keywords and search strings were used to search the reference databases and other sources:

  • “classroom techniques” + mathematics
  • “instructional effectiveness + mathematics”
  • “mathematical concepts”
  • “mathematical instructional practices”
  • “mathematics curriculum”
  • “mathematics instruction”
  • “mathematics instruction” + “multiple methods”
  • “mathematics instruction” + “standard algorithm”

Databases and Resources

REL Central searched ERIC for relevant references. ERIC is a free online library, sponsored by the Institute of Education Sciences, of over 1.6 million citations of education research. Additionally, we searched Google Scholar and Google.

Reference Search and Selection Criteria

When searching for and reviewing references, REL Central considered the following criteria:

  • Date of the Publication: The search and review included references published between 2010 and 2020.
  • Search Priorities of Reference Sources: Search priority was given to ERIC, followed by Google Scholar and Google.
  • Methodology: The following methodological priorities/considerations were used in the review and selection of the references: (a) study types, such as randomized controlled trials, quasi-experiments, surveys, descriptive analyses, and literature reviews; and (b) target population and sample.

This memorandum is one in a series of quick-turnaround responses to specific questions posed by educational stakeholders in the Central Region (Colorado, Kansas, Missouri, Nebraska, North Dakota, South Dakota, Wyoming), which is served by the Regional Educational Laboratory Central at Marzano Research. This memorandum was prepared by REL Central under a contract with the U.S. Department of Education’s Institute of Education Sciences (IES), Contract ED-IES-17-C-0005, administered by Marzano Research. Its content does not necessarily reflect the views or policies of IES or the U.S. Department of Education nor does mention of trade names, commercial products, or organizations imply endorsement by the U.S. Government.