“Low Floor, High Ceiling” Problems
“Low Floor, High Ceiling” math problems have multiple entry points so they are accessible to all students, but they can also be solved at higher levels. These rich problems have the following characteristics:
- Require an inquiry approach when solving.
- Do not have a predetermined solution pathway in advance.
- Have many possible representations of solutions and strategies.
- Involve the process of exploring and understanding the deeper nature of mathematical concepts as a by-product of solving the problem.
Is It Still a Triangle if it’s Not Pointing Upward?
Triangle is rotated to expand students’ concept of a triangle.
When young children are first introduced to shape categories, like the triangle, they are presented with symmetrical models such as an equilateral triangle (all sides and angles equal in measure). Children are usually shown the equilateral triangle positioned only one way- with one vertex pointed upward, and a horizontal base.
However, research suggests that as soon as these typical examples are introduced, a variety of different positions and sizes should also be shown to children so that their notions of triangles don’t become rigid and limited to only one type of triangle, or a single example of a shape. “Children two to three years of age are not too young for this type of learning” (NCTM, 2010).
“There’s lots of ways you can make a triangle”
Examining various triangles and determining whether or not they are, in fact triangles, leads children to grapple about specific attributes that define what a triangle is (three sides, three corners, straight lines, etc.). As children experience a broader variety of each shape, they begin to build more accurate geometric concepts and ideas. Through this investigation process, the Fours are laying the foundation for geometric discussions that will take place in later grades, such as transformations (rotations, flips, slides), symmetry, and angles.
Observe the Fours as they explore and debate what it is that makes a triangle a triangle, and in the process, redefine their ideas of triangles.
What can a student in the fours do when they run out of a certain sized block?
They can use the “recipe” or conversion chart they made to create the block size they need from other blocks.
Students in Pre-K used classroom blocks to estimate, and then test, how many of each of the same size smaller blocks it would take to cover one double unit block. The class worked in pairs on this investigation, and then the class created posters of their mathematical conversions. They have found this conversion chart, or “recipe” useful to refer to when they run out of a certain block size because they can now create the size they need by combining (composing) other blocks. Continue reading
Do multiplication arrays need to be introduced as a ready-made convention, or can they be “constructed” by children using concrete objects while engaged in an open-ended task?
(An action research collaborative effort by second grade lead teachers, Tasha Hernandez, Bill Miller, and Lower School Math Coordinator, Debra Rawlins)
We began this project as a way to build math relevance into the annual second grade canned food drive. The students visit Saint John’s Food Pantry and spend a morning learning about the needs of the community. Then they pitch in and help out the staff for a few hours. When they return to school, they create signs advertising the canned food drive, and place collection boxes at various locations around the school.
Teachers posed this question to first graders during their study of geometry.
First graders went beyond naming basic two-dimensional shapes to exploring specific characteristics, or attributes, that define them. Their conversations evolved from seeing a shape as a “whole” such as a triangle, to analyzing and deciding the specific features that prove that it is a triangle. Students decided that a shape can only be called a triangle if it has three straight lines that make up the sides, three corners, or vertices, and does not have any openings (it needs to be closed). Through this specific definition, they discovered that there are many types of triangles. They used this critical thinking foundation to explore a variety of two-dimensional shapes.
“Does a Circle Have Sides?”
This was an interesting question to ponder. First grade teachers, Sarah and Ariane, posed it to their students to see if they could apply the skills they had learned about defining geometric attributes to this question. It turns out that it wasn’t an easy question to answer! Continue reading
Math with Social Justice Relevance
It was a chilly day in January.
Fourth grade students and teachers went downtown to walk the streets of the busy financial district to meet food cart vendors. They were able to observe the variety of international selections that vendors were selling, and sample the delicious-smelling food.
Over 90% of the food vendors in New York City are first generation, or recent immigrants. This field trip gave students the opportunity to talk directly to the people who stood inside these carts, cooking food that reflected the cuisine of their home countries. Students were curious to hear the stories of where they immigrated from, and how they happened to enter into the business of selling food on the street. Continue reading
Third Graders Explore Area and Perimeter by Measuring a “Pocket Park”
As part of their study of area and perimeter, third graders in Elaine and Jessie’s class measured the perimeter of “Little Red Square”, the small pocket park that lies just in front of LREI on Sixth Avenue. Each class divided into small groups and used trundle wheels to measure the four sides of the park. Then they calculated the perimeter by adding up the side dimensions. When the class looked at the set of data, they realized that their perimeter data varied, and they attributed this to the inexactness of using the trundle wheel. They decided to use the middle number of the data set (the median) as their “working” perimeter for the park. Continue reading
How many New York City blocks is it to the Apollo Theater from LREI?
(The Apollo Theater is located at 125th Street in Harlem)
This problem seemed easy enough until Tasha’s second grade realized that the West Village, where LREI is located, isn’t laid out in an organized city grid system, like the rest of Manhattan is. An interesting math problem ensued, and the class enlisted Nick, LREI’s resident historian to help us understand why the streets in the West Village are so confusing! Continue reading