Math instruction has changed drastically over the years from rote practice and memorization as the primary methodology, to instructional strategies that promote a more conceptual understanding of math.
For example, math instruction based on the Common Core Standards focuses on the application and connections of the concepts being taught, as well as fluency in order to help students attain a deeper understanding. Students are encouraged to think more like real mathematicians and scientists, for example, whereas in the past the primary goal was to get the answers right on the quiz or test.
This deeper dive into learning math can help students navigate the complexities of life in the 21st century, and prepare them for jobs that will, almost inevitably, no matter the field, rely on technologies that we may not even know about yet. What will our 10-year-old 5th graders need 15 years from now, when they’re in the working world? That’s one of the challenges of teaching in today’s lightning fast and ever-changing technological landscape.
Since first formally introducing the learning strategy of “interleaving” into my middle school math class earlier this year, I've been continuing to think about how interleaving can help both teachers and students, alike. How can we ensure that our students are prepared for their reality 10, 20 or 30 years from now? What tools can we teach that will help them hone their problem-solving skills, for instance?
This deeper dive into learning math can help students navigate the complexities of life in the 21st century.”
I recently had a great conversation with Dr. Megan Sumeracki, a learning science researcher and Assistant Professor at Rhode Island College, which led to a sort of epiphany about the nature of math instruction, especially how it relates to what mathematics instruction entails beyond simple rote and memorization. The mathematical practices, which are now the basis for most common core math instruction are based equally around conceptual understanding, application, and fluency. The goal is to ensure that students learn deeply. Enter learning science principles, like interleaving.
To explain, interleaving is a science of learning strategy that encourages teachers to “mix up” topics during their instructional sessions. For example, typically we teach in “blocks” like studying concept or skill A, then B, then C. When teaching this way, we prompt the students to learn topic A and give a lot of practice on A before moving on to topic B, and so on.
Interleaving promotes a less predictable pattern like A, C, B, C, A, B. With interleaving, the students are practicing the various topics in a mixed up fashion, so that they are often switching back and forth among topics during any given practice session. Interleaving helps learners retain information and learn new skills in math by forcing them to analyze the differences in the structure of the questions and to correctly apply the appropriate problem-solving strategy to the question rather than blindly completing similar problems.
Sumeracki, and I were reviewing lesson plans, looking for learning science principles and it dawned on me that old-school math instruction was often about memorizing math in order to develop fluency. And this requires massive amounts of practice, which is why so much of math instruction of the past involved memorization of rules and rote exercises, lovingly called “drill and kill.” Very rarely did we go beyond the simple memorization of math facts and rules. In today’s world, we can, and must do better for our students by deliberately teaching them learning strategies, within the context of our teaching, as well as within the context of their lives.
...so much of math instruction of the past involved memorization of rules and rote exercises....”
Interleaving can sometimes feel counter intuitive since it pushes people beyond rote practice. Doesn’t it make more sense to practice one thing, over and over again, until we have it down - then, move onto the next skill or concept? But, research tells us otherwise. The practice of mixing things up, or interleaving, facilitates longer-term retention and mastery because students are forced to truly understand the math in order to answer the questions correctly. They have to learn both how to solve a given problem, but more importantly, perhaps, when to apply which principles.
In conversations with teachers from many different schools, a great number have shared that they have stopped giving more than 20 questions for homework, instead choosing five or six meaningful and “deep” questions or problems. These homework assignments tend to be conceptual or application-based questions rather than just procedural, or more tied to rote learning. This creates interleaving or at least a similar effect in that it removes the repetitive practice problems and replaces them with questions that highlight the differences in structure and keeps students on their toes because they have to analyze the nature of each problem.
For students to truly recognize the differences between problems and use those subtle differences to learn deeply, requires the specific teaching of these learning strategies and concepts. Strong students will automatically look for structural differences or contextualize the numbers in order to make sense of them. Struggling students, however, are more likely to just answer the questions incorrectly and not know if and/or why the answer is correct or incorrect. Strategic and almost instant feedback is critical for these students to know that they are wrong, so that they can be guided through the process of noticing the nuances of each question.
This specific skill of analyzing question types or details needs to be taught in the classroom, not just reserved for homework review procedures. Hopefully, students will then transfer those skills to their independent work. These are critical learning skills, and what better gift can we give our students than the ability to learn in new ways that can open doors that might otherwise have remained closed tight for struggling students?
Dr. Kim Kelly is one of the teachers participating in the Overdeck Foundation’s “Science of Learning Video Project,” coordinated in-part by The Learning Agency. Kelley is a math teacher at Sky View Middle School in Leominster, Massachusetts. She recently completed her Ph.D. in Learning Sciences and Technology at Worcester Polytechnic Institute.
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