By Dr. Mona Tauber, Math Specialist at The Langley School
What It Means to “Really” Answer a Math Problem: A deep dive into the math skills we should expect from our students
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Category Archives: Math
Have You Ever Heard of the Math Ceiling Fairy?
By Inga Schoenbrun and Janice Graves, Math Specialists
Have you ever heard of the Math Ceiling Fairy? Many students’ eyes are glued to the ceiling anxiously awaiting the fairy’s appearance to give them the answer to math problems like 8×7. The fairy may be a figment of our imagination, but reliance on memorization is not. A recent article in Scientific American details research from Stanford University showing that an emphasis on memorization, rote procedures, and speed impairs learning and achievement in math.
At Langley, we strive to develop strategies that allow our students to use what they do know to figure out something they don’t. In the example of 8×7, a student might employ the double strategy: I know 4×7 is 28 and 8 groups is double 4 groups, so 8×7 is double 28 or 56! This student understands not only how to double numbers, but also the structure of multiplication as equal groups. Another student might realize they know 7×7 is 49, and simply add one more group of 7 to get 56.
Students who recognize the effect operations (addition, subtraction, multiplication, and division) have on numbers are more adept at solving complex, multi-step problems. They construct concrete or pictoral models to illustrate the problems and allow themselves to take risks to solve problems using a variety of methods. Strategy-based learning leads to generative knowledge where memorization lends itself to temporal and compartmentalized learning.
Do we value math fact fluency? Absolutely we do. We are committed to building fluency hand in hand with number sense and mental math strategies. Listen closely as your child computes numbers and marvel at his or her creativity and efficiency. It just might surprise you.
Math and the 21st Century Learner
By Beth Morris, Math Resource Teacher
Our world has been transformed dramatically over the last few years, and so has the way in which we teach math. No one really knows what the lives of the next generation will look like. So how can we best prepare our children for a future that is largely unknown? We need to teach them how to think. Thinking is universal and will transcend any amount of innovation to come. If children know how to think and reason logically, then they will be able to adapt in a world that is rapidly changing.
Recent mathematics reforms call for a much different approach to teaching math in order to meet the needs of the 21st century learner. Students are exploring the math that they are learning, testing their beliefs, grappling with tough questions, and reflecting on their thought processes. They are collaborating and communicating with each other and exchanging ideas. Teachers are guiding their students to refine their thinking and to make connections between concepts and ideas. The goal is for children to build a deeper understanding of the math that they are learning that is useful for them now, but also in the future.
Conceptual understanding is now a major focus in math. Students are not just learning procedures. In fact, research shows that when procedures are introduced too early, children lose their curiosity about numbers and their enthusiasm for learning math. Instead, students are digging deeper and investigating why and how procedures work before those procedures are formally taught. They are being exposed to various strategies for solving a problem and discussing which of those strategies is most efficient and effective. They are modeling problems and using other tools to see those strategies in action. When math is taught like this, children are given the opportunity to make sense of the math that surrounds them. They are thinking like mathematicians.
Mathematicians use estimation, look for patterns, and utilize mental math strategies when solving problems. This is exactly how we teach our students here at Langley to think. For example, we encourage our students to look for number relationships. Children often learn their doubles addition facts first, and they should recognize 7+7 and 6+8 as related facts. Using manipulatives, our students learn that by taking one away from the first addend and giving it to the other that the sum remains the same.
Our students are also thinking about “friendly numbers.” Multiples of 10 and 100 are easier to work with. When subtracting 98 from 276, our students might start by subtracting 100 and then adjust their thinking accordingly. We urge our students to think about numbers in a variety of ways. For instance, 564 can be 560+4, which is helpful when adding 126. Or 564 can be 400+120+44, which is helpful when dividing by 4. This sort of flexibility with numbers leads to learning procedures with authentic understanding, makes computation much simpler, and lays the foundation for future success in algebra and beyond.
For more information about math instruction at Langley, current parents may view the presentation from the September 30 Math Curriculum Coffee by clicking here and logging in to our CampusNet site. As always, parents are also welcome to approach their child’s teacher or division head with any questions or concerns.