## Beginning of article

A mathematician's work involves observing something in the natural world, making an abstraction, recording a finding, and then communicating the outcome using the language of mathematics (Devlin 1994). Mathematicians connect their observations with what they already know and attempt to fit it into some mathematical pattern (Polya 1957).

In the last hundred years the type of work involved in the study of mathematics has undergone a change. Mathematics has grown to include so many new and different facets that mathematicians have had to redefine their own work. Instead of defining the field as a collection of isolated disciplines, the focus of today's mathematicians is on seeing patterns and making connections within mathematics (Devlin 1994).

This article describes two teachers, Rachael and Tamara, whose own understanding of the work of mathematics has changed their approach to teaching mathematics. Rachael and Tamara, along with a colleague, formed a study group to research the investigative process used by today's mathematicians. Their synthesis of that process--observing, abstracting, recording, and communicating--serves as a model for students in their classrooms.

Modeling the Mathematics Process

Rachael's first-grade students are gathered on the floor in front of her easel. The focus for this week's mathematics investigation is geometry. Rachael starts her demonstration by making a picture with pattern blocks (see fig. 1).

Rachael shares her thinking with her students, "I used different shapes to make this picture of a tree. I wonder, how many shapes do I have here? I have two different kinds of shapes. I used these trapezoids at the bottom to form the trunk. How many are there? Let me count them: 1, 2, 3." By talking aloud, Rachael is showing students how she thinks mathematically about what she sees.

"Now I am going to write the number 3 down. Let me count the shapes at the top. I see these are squares. There are four: 1, 2, 3, 4. I am going to write that number down. I'm wondering how many shapes there are altogether. Let's see [ldots] there are three trapezoids and four squares. I'll start with the four squares and count the rest: 5, 6, 7. Now I'm going to record that number. There are seven shapes altogether." Rachael's chart paper shows the numbers 3, 4, and 7. She knows that the purpose of the demonstration is to model the mathematical process. Geometry provides the context to teach this process.

"Since we're studying geometry, and geometry is the study of shapes, your investigation needs to involve shapes. I wonder what you might do." Students then go back to their seats to start their own investigations.

Young Mathematicians at Work

As the class is dismissed from the easel, Rachael roves to make sure that the children know what they will investigate. Students are either beginning a new geometry investigation or completing one from the day before. Samantha has used the shape tracer to make a repeating pattern and recorded 10 for the number of shapes she has used. Jesse has made a symmetrical pattern and has observed the number of sides on the shapes in her picture. She has recorded "3 + 3 + 3 + 3 + 3 + 3 = 18."

An investigation begins with the student making an observation within a given focus. The student then attempts to describe the observation using mathematical language and records the description in his or her math journal. During a conference, the teacher uses the student's recording to make a teaching point. Then the student is ready to share his or her findings.

Observation and abstraction

Mathematicians begin their work by making observations. Then they connect their observations with what they already know. Very often, the observation fits into some mathematical pattern (Devlin 1994). Creating a mathematical relationship from what is observed externally is the act of constructing …