Dr. Matt Larson
The roots of traditional teaching mathematics instruction run deep in American history: the first true American school mathematics textbook, Arithmetic, was published in 1788 by Nicolas Pike, who recommended that teachers follow the following script when teaching mathematics:
State a rule.
Demonstrate the rule with an example.
Have students practice the rule.
Pike’s approach to mathematics instruction casts a long shadow; it is still the dominant approach in many classrooms across the country. If Pike’s method is outdated however, the question remains: what should mathematics instruction look like today? I believe that effective math teaching today shares five critical features:
1. Students Develop Conceptual Understanding and Procedural skills
The history of mathematics education in the United States can be seen as a pendulum swing between an overemphasis on procedural skills at one end, and an overemphasis on conceptual understanding at the other. However, the research is clear: students need both. Students should not only learn how to do something, but also why it works and when to apply their knowledge and skills.
2. Students Communicate with Peers About Mathematics
More traditional mathematics classrooms are often quiet. Today, effective classrooms are filled with purposeful discussions among classmates and the teacher about concepts, skills, and problem solving strategies. Students who communicate with their peers and teachers about their thinking, solution pathways, and insights into how they solved a problem develop a deeper understanding of math. One way for educators to generate discussions is by asking open-ended follow-up questions such as, “Why?” “Can you explain?” And “How do you know?” after students answer questions.
3. Students Develop Perseverance and Practice Mathematics
Anyone who becomes an expert at an activity must practice. Learning mathematics is no different. In effective classrooms, students practice mathematics both in class under the guidance of the teacher and at home, independently. In order to be successful, students also need to practice problems that require them to reason, problem solve, and develop the confidence to know they can solve problems they have never seen before. Teachers in effective classrooms resist the urge to water down expectations in an effort to make work easier for students; instead, they expect them to try and persevere, particularly when it’s hard.
4. Students Use Teacher and Peer Feedback to Learn from Mistakes
When learning mathematics, students should make an effort to solve problems, embrace the possibility of making mistakes, and rethink their solution pathway. Mistakes should not be viewed as a setback, but an opportunity to learn, which means the most effective mathematics instruction leverages meaningful feedback and student action on that feedback as part of a formative learning process. In effective classrooms, homework does not count for the bulk of a student’s grade because students are encouraged to get feedback on their homework and view any mistakes made as a learning opportunity.
5. Students Use Technology to Support Learning
Technology, including computers and graphing calculators, should be used to help students develop a deeper understanding of mathematics. These tools permit students to see visual representations of mathematical ideas, receive immediate feedback on their work, have access to 24/7 tutorials, and collaborate with peers on homework and projects. Although technology does not and cannot replace an effective classroom teacher, its presence is a critical component of effective mathematics classrooms today.
Although the above are certainly not the only features of effective mathematics classrooms today, I believe they are critical features that, when in place, will better serve student learning today. We are seeing movement away from Pike’s outdated method and towards a more engaging and meaningful teaching method, and these five critical features are a step in the right direction.
To learn more about Nicolas Pike and the history of math instruction in the United States, please refer to Balancing the Equation: A Guide to School Mathematics for Educators and Parents (2016).