Author Archives: drawlins

Controversial Crowd Counting

  By drawlins • May 19, 2017

“Almost everyone who has tried to make a crowd estimate has a vested interest in what the outcome of the estimate is”

–Charles Seife, Professor of Journalism and Mathematician at NYU.

4th Graders use the Jacob’s Method of Crowd Counting to Estimate the Protest Crowd at Washington Square Park on Wednesday, January 25th, 2017. 

On Wednesday evening, January 25th, 2017, a protest crowd gathered at Washington Square Park in response to President Trump’s executive orders on immigration that he signed earlier that day. The New York Post reported that “Hundreds Gathered” (Marino & Perez, 1/25/17). The New York Observer (Toure, 1/25/17), called it a “Massive Demonstration”. Vogue (Codinha, 1/26/17), reported that “Thousands of New Yorkers Gathered”. So just how many people showed up for this rally? LREI fourth graders wanted to find out.

The Jacob’s Method of Crowd Counting

Herbert Jacobs, a University of California journalism professor in the 1960’s, devised a basic density rule that has been widely accepted. Watching students protesting the Vietnam War from his office window, Jacobs saw that they had gathered on a plaza that was arranged in a grid. He counted those in a few squares to get an average number per square and multiplied that by the total number of squares. He also came up with a basic density rule that states a “light crowd” has one person per square meter, and doubled that for a “dense crowd”. A “heavy crowd” would have as many as four people per square meter, according to this method.

From Cooking to Counting

  By drawlins • May 9, 2017 , ,

First Graders Cook, Question, and Count

By Julie Kim, First Grade Associate teacher

In the real world, we confront daily math problems through a process of noticing and wondering. After our mind has determined a question about a scenario, whether it is counting how many more blocks you need to walk or how many servings to cook for dinner, we proceed to the next step: plan, search, and gather. We plan for what steps we are going to take in order to answer the question. We search for the separate variables and pieces of information that we need in order to solve the question. We gather these pieces of information then puzzle them up in a way that will help us solve the problem. Will we, as mathematicians, get the answer we are looking for the first time around? Not always. Will we get an answer immediately? Not guaranteed. Will we persevere and try over and over again until we do? We should. Through real world work, first graders develop stamina and perseverance as they attack the challenges and questions that they are eager to solve. This is where the real work happens.

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Expanding Geometry Beyond the Equilateral Triangle

  By drawlins • March 3, 2017 , ,

Is It Still a Triangle if it’s Not Pointing Upward? 

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. 

The Math Relationship of Blocks

  By drawlins • February 26, 2017 , , ,

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

Cooking Up Arrays

  By drawlins • January 31, 2017 , ,

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.

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LREI Math Philosophy

  By drawlins • January 31, 2017

A Summary of LREI’s Math Philosophy Written by Teachers from Lower, Middle, and High School: 

Children at every developmental level deeply engage in the discovery of mathematics through the inquiry and investigation of interesting problems. LREI students collaborate on solving worthwhile problems that have rich mathematical underpinnings. Problems can be solved on a variety of levels, have multiple entry points, and therefore provide both accessibility and challenge to the abilities of all students in a class.

A good problem is also defined as not having a predetermined solution pathway that is known in advance. A worthwhile task, or problem, will often open the door to other interesting conjectures and theories that students have, and support an emerging mathematical habit of mind.

Throughout each unit of study, students develop and maintain computational fluency. Students are instilled with an enthusiasm for mathematical challenges and the open-endedness of mathematical inquiry. Our students develop perseverance and a sense of ownership of mathematics. Students achieve genuine understanding when they actively construct knowledge rather than passively absorb; when they use it to solve problems rather than memorize facts and formulae.

Students throughout the divisions learn through inquiry and collaboration. Emphasis is placed on the presentation and communication of mathematics in written, oral, graphic and symbolic forms. Students are encouraged to clearly describe, explain and support their findings with valid evidence.

In the Lower School, students display their work using a document camera and engage their peers in the discussion of a problem. In the Middle School, students can share their thinking by reflecting work done on an iPad or graphing calculator to a Smartboard. Working in small groups is a commonly used practice across the grades. In High School, students put their mathematical ideas on small whiteboards and then each group presents to the rest of the class, and the classmates ask questions.

Consistent with LREI’s philosophy, the mathematics department values creativity and critical thinking, as well as risk-taking when problem solving as a means to confront life’s challenges and variables. Our students understand that mathematics has the power to model the world around us on multiple levels. Projects and problems, offered in math classes, have relevance to students and their lives.

How Many Sides Does A Circle Have?

  By drawlins • January 30, 2017 , , ,

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