What evidence-based mathematics practices can teachers employ?
Page 8: Effective Classroom Practices
A number of other classroom practices are supported by moderate levels of evidence, even if they have not yet met the requirements to be considered evidence-based. Implementing these kinds of effective practices in conjunction with an EBP is yet another way teachers can improve their students’ mathematical understanding. Among these effective classroom practices is:
- Encouraging student discussion
- Presenting and comparing multiple solutions
- Assessing student understanding
- Students’ mathematics achievement improved significantly when student discussion was an integral part of instruction.
(Ing, et al., 2015; Huinker, 1992)
- When teachers presented multiple solution strategies for solving the same problem, students demonstrated significant increases in procedural flexibility, conceptual knowledge, and procedural knowledge.
(Durkin, Star, & Rittle-Johnson, 2017; Jitendra et al., 2011)
- More than 30 years of research indicates that curriculum-based measurement (CBM), provides stable, accurate student screening and progress monitoring data in mathematics.
(Lembke & Stecker, 2007; Tindal, 2013)
- Teachers have successfully used error analysis to identify the problem-solving difficulties and conceptual errors made by their students.
(Kingsdorf & Krawec, 2014)
For Your Information
When implementing the effective practices listed above, teachers will expect students to explore new concepts, attempt challenging problems, discuss their thought processes, or be open to corrective feedback. However, many students might not feel comfortable engaging in these activities, therefore, teachers need to establish a supportive and safe classroom environment. Within this type of environment, teachers can stress that making mistakes is not only acceptable but also valuable because doing so creates opportunities to identify and address faulty thinking or misconceptions.
Encouraging Student Discussion
How does this practice align?
CCSSM: Standards for Mathematical Practice
- MP3: Construct viable arguments and critique the reasoning of others
Student discussion or discourse is a practice that encourages students to express their mathematical reasoning. It allows them to become aware both of their own problem-solving processes as well as those of others, and to refine their conceptual understanding. Additionally, student discussion allows the teacher to assess student understanding. This practice could be implemented during whole-group discussion or during small-group activities. To implement this practice, teachers should:
- Establish discussion procedures (e.g., students justify answers by explaining their reasoning, students ask other students for clarification).
- Establish behavioral expectations (e.g., respect others while they are talking).
- Provide supports for students with disabilities (e.g., word wall of mathematics vocabulary, opportunities to discuss thinking with a partner before sharing in a whole-group).
- Create a list of prompts to encourage discussion among the students (e.g., “What do you think about Shay’s explanation?” “Can you add to Ramsee’s explanation?”).
- Provide sufficient wait time so that students have an opportunity to formulate a response.
The video below depicts a teacher encouraging his students, during whole-group instruction, to discuss their thoughts and ideas about a series of problems they have solved regarding a V-pattern. As you watch, note that the students have difficulty articulating how they arrived at their answer, but the teacher continues to prompt and guide the discussion (time: 3:07).
Presenting and Comparing Multiple Solution Strategies
How does this practice align?
According to CCSSM, comparing multiple solution strategies allows children to gain an understanding of the relationship between:
- Addition and subtraction
- Multiplication and division
Teaching multiple ways to solve a problem helps students to develop flexibility (i.e., understanding that a problem may be solved accurately using different procedures and being able to use efficient procedures) and might support conceptual understanding of the procedure. To do this, teachers should:
- Demonstrate how to solve a mathematics problem using multiple strategies.
- Present the strategies side-by-side.
- Guide students through a process of comparing multiple strategies to solve a problem.
- Use common labels to draw attention to similarities.
- Prompt for specific comparisons tailored to your learning goals.
- Be sure that students, not just the teacher, are comparing and explaining.
- Include a summary of the main idea from the comparison, highlighting key points.
- Reinforce the concept of being able to solve a problem using multiple strategies.
- Encourage students to solve problems using a strategy of their choice.
- Ask students to share their strategy with peers in a small-group or whole-group setting. By doing this, students have opportunities to see how other students solved the problem, which increases their exposure to multiple solution strategies.
Note: This does not mean that each student must solve every problem using multiple strategies, a common misinterpretation of the CCSSM requirements. Rather, students exposed to multiple strategies have a greater possibility of finding at least one problem-solving approach that they can understand and apply.
Watch the video below for an example of how a teacher can present and compare multiple strategies for solving a two-digit addition problem (time: 4:31).
Assessing Student Understanding
How does this practice align?
High-Leverage Practice (HLP)
- HLP6: Use student assessment data, analyze instructional practices, and make necessary adjustments that improve student outcomes.
As we previously explained, assessing student understanding allows teachers to determine whether students have learned the mathematical procedures or concepts covered in class. Teachers can use different types of assessment data, including formative assessment and error analysis, to make instructional decisions (e.g., identifying what they need to revisit or reteach).
Formative assessment is the ongoing evaluation of student learning as a means of providing continual feedback about performance to both learners and instructors. By using formative assessment teachers can determine what students have mastered and what concepts they are struggling with. Teachers can use both informal and formal formative assessments. Informal assessments include exit tickets, quizzes, and class work samples. Formal formative assessments include curriculum-based measurement (CBM), sometimes referred to as general outcome measures (GOM), which is a type of progress monitoring.
For more information on CBM, view the following IRIS Module:
Error analysis is the process by which instructors identify the types of errors made by students when working mathematical problems. It allows teachers to assess student understanding, or misunderstanding, and to identify and analyze the student’s error patterns—errors a student repeatedly makes when solving a mathematical problem. The teacher can use the information from the error analysis to target instruction to help the student understand the correct procedure for solving the problem. If the reasons for the student’s incorrect answers are not apparent, the teacher can ask the student to describe the procedure she used to solve the problem, as is illustrated in the box below.
Diane Pedrotty Bryant, PhD
Project Director, Mathematics Institute for
Learning Disabilities and Difficulties
University of Texas at Austin
To learn more about error analysis, visit the following IRIS Case Study Unit: