Helping Mathematics Students Reach Deep Understanding

It comes as no surprise that what teachers do in the classroom impacts students' learning of mathematics. During a single class period, a teacher makes countless decisions about how to present material, structure tasks, and guide student learning. Some decisions seem large, such as how to grade students and how to introduce new concepts. Other teacher decisions may seem less significant, such as the choice of how to position students' seats, which problems to assign for homework, or whether or not to ask students to raise hands before answering questions. In essence, effective teaching is effective decision-making, as the collection of big and small decisions made by mathematics teachers makes a tremendous difference in students' learning. How do teachers decide what to do when they teach? These decisions may be based on their training or professional development experiences, what they experienced as students, or what they have learned from colleagues. A recent practice guide titled Organizing Instruction and Study to Improve Student Learning aims to supplement and inform teachers' instincts and experiences by identifying research-based instructional strategies that teachers of all content areas can use to improve student learning.

The practice guide identifies the following seven recommendations (Pashler et al., 2007, p. 2):

1. "Space learning over time. Arrange to review key elements of course content after a delay of several weeks to several months after initial presentation."
2. "Interleave worked example solutions with problem-solving exercises. Have students alternate reading already worked solutions and trying to solve problems on their own."
3. "Combine graphics with verbal descriptions. Combine graphical presentations (e.g., graphs, figures) that illustrate key processes and procedures with verbal descriptions."
4. "Connect and integrate abstract and concrete representations of concepts. Connect and integrate abstract representations of a concept with concrete representations of the same concept."
5. "Use quizzing to promote learning. Use quizzing with active retrieval of information at all phases of the learning process to exploit the ability of retrieval directly to facilitate long-lasting memory traces."
   
   5a. "Use pre-questions to introduce a new topic."
   
   5b. "Use quizzes to re-expose students to key content."
6. "Help students allocate study time efficiently. Assist students in identifying what material they know well, and what needs further study, by teaching children how to judge what they have learned."
   
   6a. "Teach students how to use delayed judgments of learning to identify content that needs further study."
   
   6b. "Use tests and quizzes to identify content that needs to be learned."
7. "Ask deep explanatory questions. Use instructional prompts that encourage students to pose and answer deep-level questions on course material. These questions enable students to respond with explanations and support ... deep understanding of taught material."