Math Takes Time
By Cathy Seeley, Mathematics Education Expert
How much time does it take for students to explore a mathematical situation, understand a mathematical concept, solve a mathematical puzzle or problem, or learn a mathematical procedure? That depends.
We know from decades of research how teachers can more effectively support all students in understanding mathematical concepts, becoming fluent in mathematical procedures, and developing as mathematical thinkers and problem solvers. Much of this research has been summarized by the National Council of Teachers of Mathematics (NCTM) in their 2014 book, Principles to Actions and used as a basis for the eight Effective Teaching Practices presented in that book (NCTM, 2014). Consistent with other recent recommendations, those practices describe classrooms based on clear mathematical goals where students openly discuss their thinking about mathematical problems as part of their journey toward mathematical understanding. They productively struggle with appropriate levels of rich tasks that build on what they know to challenge their thinking and help them learn with lasting understanding. This kind of teaching prepares students to be able to tackle all kinds of problems they may encounter not only in math class but in other content areas and in their lives outside of school.
Lessons based on this kind of student-centered teaching take more time than directly presenting content to students and asking them to remember and apply it. Recognizing the benefits of extended time for students to explore ideas and engage in constructive discussion, some schools have implemented longer class periods than a traditional 45-minute period. But even if you don’t have the luxury of longer class periods, students can still reap the benefits of extended time when a teacher carries over instructional lessons to two or more class periods. Teachers can learn to bring closure to one day’s work as they set up continued related work for the next day. During each class period of the lesson, students wrestle with mathematical ideas, discuss their ideas with each other, and justify their reasoning to their classmates and their teacher. Such rich thinking and discussion are far less likely to happen in a single-class lesson where a teacher tries to set up a lesson, present material to students, give a few practice exercises, and bring closure to the lesson.
Benefits of Longer Lessons
If we let go of the idea that a math lesson begins and ends in a single class period, we open the door to all kinds of opportunities for students. With longer math lessons, students have time to Be Curious, Discuss, Reason, Solve Problems, and Make Connections.
Students have time to BE CURIOUS.
We want students to build on their natural curiosity about mathematical ideas. We also want them to develop a deep conceptual understanding of — and proficiency in — mathematics. But accomplishing those goals takes time, and calls for a restructuring of how we conduct math class. It involves opening lessons with a compelling question, puzzle, or problem and giving students space — and time — to wonder about it, to ask questions about it, and to dig into the ideas that lie within the task. Creating a classroom full of curious students genuinely engaging in math talk and math thinking allows students to find a personal connection to math and engage in sense-making at their own pace.
Students have time to DISCUSS.
The role and importance of classroom discourse appear over and over again in NCTM’s Effective Teaching Practices. Constructive discourse among students and between students and the teacher calls for more time than is normally available in 45 minutes. Students need time to talk about math with each other, with their teacher, and in whole-class discussions. They need to be able to verbalize their reasoning and explain how they came to an answer or what they may have tried as they grappled with a concept. And teachers need time to listen and respond to that discussion. Such rich thinking, discussion, and teacher feedback are far less likely to happen in a single class session with limited time than in a longer lesson, whether in a single longer class period or over two or more days.
Students have time to REASON and SOLVE PROBLEMS.
Developing reasoning and problem-solving skills is an essential foundation for students to deal with the wide range of problems they will face in school and in their rapidly changing world outside of school. They build those skills by wrestling with increasingly complex mathematical concepts, problems, and ideas.
Math doesn’t present itself to students in neatly stated word problems; math comes at students disguised as real life.
Dealing with problems they have never seen before requires that students encounter problems in school they haven’t already been taught how to solve. Dealing with an unexpected or unknown problem is far different from merely solving a word problem that applies the procedure just learned. It’s only when students have time to grapple with ideas they may not even recognize as mathematics, that their understanding — and ownership — of those ideas evolves and deepens. They need time to engage in productive struggle, build on what they already know to create new knowledge, and move beyond short-term or shallow learning toward knowledge and understanding that lasts. Math is more than memorization — our lesson organization should be designed around what we want our students to take with them after the lesson is over. If we want them to come away with the ability to think, reason, and solve problems, we need to give them time to do just that in math class.
Students have time to MAKE CONNECTIONS.
Offering more time also allows students to build connections among mathematical ideas — “What’s the same and what’s different between this problem involving money and the work you did a while back on similarity in geometry?” If we organize our content into larger ‘chunks’ rather than a long list of smaller ‘bits’, we can structure longer time to help students connect the smaller bits within those chunks. We can also provide time for students to build connections between mathematics and the world outside of mathematics.
Exploring mathematical ideas, concepts and problems involves all of the aspects described here. But all of that depends on a teacher helping students connect their wrestling, struggling, discussing, and justifying to a mathematical output — the mathematical goal of the lesson. Perhaps the most important role of the teacher is to help students make the connection between what they’ve been doing and the mathematical concept or procedure that is the goal of the lesson. Sometimes in the process, they might even gain additional understanding beyond the goal. However, taking the time to pull the pieces together with a mathematical outcome is a critical component of an effective lesson. Longer lessons allow a teacher to bring closure to the lesson by helping students make those important connections.
The Bottom Line
The issue isn’t about designing longer lessons. It’s about creating engaging lessons that tap into students’ curiosity and call for students to think and discuss, not just listen and watch. It’s about having enough time for students to wrestle with ideas and engage in conversation with each other, thinking out loud about possible approaches to a problem and justifying their thinking, perhaps even correcting and modifying their own thinking as they do it. It’s about allowing enough time for students to build a mathematical toolbox based on their Aha!s. It may seem like a luxury to allow students more time to deal with bigger chunks of mathematics, but it’s actually an investment in their deeper, longer-lasting learning.
So how much time does all of that take? As much as we can give them. We can start by thinking in pieces longer than 45 minutes.
To learn about the structure and benefits of longer math lessons for middle school in California Reveal Math, visit mhecalifornia.com/reveal.
Learn more about extended middle school lessons in our national PreK-12 math program, Reveal Math, at revealmath.com.
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Dr. Cathy Seeley has been a teacher, a district mathematics coordinator, and state mathematics director for the Texas public schools, with a lifelong commitment to helping every student become a mathematical thinker and problem solver. From 1999 to 2001 she taught secondary mathematics in Burkina Faso as a Peace Corps volunteer and on returning served as President of NCTM. Before and after those experiences, Cathy was a Senior Fellow at the Dana Center at The University of Texas. Cathy has been recognized for her equity work on behalf of all children by NCTM, NCSM, and the Benjamin Banneker Association. Her books include Faster Isn’t Smarter — Messages About Math, Teaching, and Learning in the 21st Century, Second Edition and its partner volume Smarter Than We Think, as well as two short books co-published by ASCD, NCTM, and NCSM: Making Sense of Math and Building a Math-Positive Culture. Cathy is an advisor/author for McGraw Hill’s Reveal Math secondary textbook series and consulting co-editor for a collection of resources by Solution Tree entitled “Growing the Mathematician in Every Student.” She continues to advocate for helping every student learn to love math and think mathematically.
References
National Council of Teachers of Mathematics (2014). Principles to Actions: Ensuring Mathematical Success for All. Reston, VA, National Council of Teachers of Mathematics.
