Making Math Make Sense

LOWER SCHOOL’S NEW MATH CURRICULUM FOCUSES ON STUDENT-CENTERED, DISCOVERY-BASED LEARNING, DRIVING HOME CONCEPTS AND DEEPENING MATHEMATICAL THINKING FOR STUDENTS.

Given a hexagon-shaped cookie, third graders in Anna Tobia’s class can tell you exactly how many friends you can split it with. Using foam blocks shaped like various polygons, they’ll show you that you can halve the cookie, and share two trapezoid-shaped pieces, or split it into three rhombuses, or even six triangles. Or, if you’re feeling hungry (and not quite so generous), you can save half of the cookie for yourself, give a third to someone else, and someone else a sixth.

While the cookie metaphor is fun, visual, and tactile, the third graders, of course, are also learning about equivalent fractions. “As they explore independently, the blocks reinforce what they know about shapes and help them connect concepts and build bridges between different kinds of math,” Tobia explained. In essence, she said, “They’re no longer doing math in isolation.”

That kind of bridge-building is exactly what Investigations 3, the Lower School’s new math curriculum, aims to do. Piloted with a geometry unit in spring with a full rollout planned for September, the goal of Investigations “is to develop mathematical thinkers with deep number sense,” said Lower School Learning Specialist Rebecca Harrison.

Through hands-on, engaging games and student-led activities, students develop both computational fluency and mathematical reasoning skills. “It gives kids agency — instead of teaching procedures that kids can use to get the right answer without knowing at all what they’re doing or what it means, Investigations uses activities that actually develop great number sense and understanding of place value,” Harrison explained. “The activities themselves are what teach girls the deeper understanding of the mathematical concepts.”

Chrissy Duffy ’13 found that discovery-based approach to be extremely successful with her first graders as they explored geometry this spring. Students start out doing some “number talk” to warm up their “math brains” before splitting into rotating workshops to make mathematical discoveries around shapes and number concepts. At one math station, students were challenged to use as many foam shapes as possible — triangles, squares, hexagons, rhombuses — to fill in an outlined design. “For some, they knew immediately that they needed to use the smallest shape to fill in the design. For others, it connected more slowly, as they learned to problem solve in this way,” Duffy explained.

As they explore independently, the blocks reinforce what they know about shapes and help them connect concepts and build bridges between different kinds of math.

Those concepts are the mathematical building blocks they’ll eventually use to discover equivalent fractions. “By the time they get to third grade, they’re using all those concepts they’re already familiar with to visualize the fact that a half, plus a sixth, plus a third, equals one. Whereas in traditional math, we’d have to use a common denominator — we couldn’t visually picture how a third and a sixth make a half,” Harrison explained.

The conceptual, rather than procedural approach also had an added benefit: increasing confidence for girls who don’t see themselves as naturally mathematically minded. “The discovery-based format makes it accessible to all,” Tobia said. “It’s not intimidating for students who don’t prefer math, and not easy peasy for students who are huge mathematicians. It provides an entry point for all students and makes math fun.”

Duffy recalled one student in her first grade class who benefited from the new curriculum. “I thought she needed more structure and procedural instruction to excel. When I started Investigations, a lightbulb went off. The fact that the lessons were tactile, visual, that she was able to use manipulatives in any way she wanted — it gave her a freedom to make her feel she could do it. She suddenly wasn’t afraid to make a mistake.”

For girls who are highly advanced in math, Investigations works because it encourages them to take charge of their own enrichment by asking “the next questions,” Harrison added. “It will stir their own curiosity to say ‘Well what does this mean? Does it work for all shapes?’ and then they’ll pursue and investigate those next questions.”

Most lessons in the curriculum include time for the class to come together and share their discoveries. “Someone sharing their discovery leads another student to say something else, and another student to say something else, and they would really build off each other,” Tobia said. “I saw them making great connections between math concepts like fractions, multiplication, and division. At times, the students were really driving the learning and I was just facilitating it.”

“There’s not just one way to get the answer,” Duffy said. “There might be five — and we encourage the girls to find the way that works for them. That’s what I love most about it. Three girls might say ‘I did it this way,’ and someone else will share a new way to reach the same answer. I was surprised at how deep the discussion would get, especially with six year olds, but each girl was excited to show her own way.”

The same applies to less obviously tactile types of math, like subtraction, multiplication, and division. Visual methods, like number lines, are employed to teach the basics of these concepts, with the goal that students will eventually develop their own algorithms for solving the problems. “Investigations asks children to look at numerous examples and then generalize a rule, and then figure out when that rule applies,” Harrison explained. “Does it apply to all numbers, or just a few?”

“We are working toward an understanding that math is flexible,” Tobia said. “It’s not 5 + 3 = 8, it can be 2 + 3 + 3 = 8. There are more discoveries, more ways to solve, and this way of teaching helps students understand that math is flexible, connected, and useful.”