Geometric Thought in Elementary School

Geometry is a part of our curriculum in Kindergarten through Fifth Grade. In Kindergarten, one of the standards reads, The student describes shapes and space, and uses basic shapes, spatial reasoning, and manipulatives to model objects in the environment and to construct more complex shapes and in Fifth Grade two of the standards read, The student describes draws, identifies, and analyzes two- and three-dimensional shapes and The student uses appropriate geometric vocabulary to describe properties and attributes of two- and three- dimensional figures (Example: obtuse & acute angles, equilateral, scalene, & isosceles triangles).

As children's' understanding of Geometry develops in elementary school, they move through three of van Hiele's five levels of Geometric Thought (Van de Walle, Teaching Student-Centered Mathematics). Visualization, Level 0, is when students recognize a shape based on its appearance. At this stage, students recognize a rectangle because it looks like a rectangle. As a student's knowledge develops and they move into Level 1, Analysis, they are able to understand classes of shapes. They don't just recognize a rectangle on its appearance, but are able to say it is a rectangle because it has four sides, four right angles, and opposite sides are parallel. Knowing the properties of the shapes gives the shape its name. When students move into Informal Deduction, Level 2, they are able to see the relationships and connections among the properties of a class of shapes. They would be able to conclude that a square is a type of rhombus because rhombi are polygons with four equal sides and opposite sides are parallel and congruent. A square fits this definition, therefore a square is a type of rhombus. Students in elementary school work through these stages, and then move into Level 3, Deduction, and Level 4, Rigor, later in their school career.

For our March Standard Snapshot, Kindergarten and Fifth Grade teachers used work from their Geometry unit to share with parents. The Kindergarten students were in the beginning of their unit and the Fifth Grade students were in the middle of their unit.

Kindergarten Student Sample

You will notice in this Kindergarten sample that the student has mastered Level 0, Visualization, because they are able to name the shape, a rhombus, based on its appearance. They have not proven, through this assignment, that they have mastered the Kindergarten standard The student describes shapes, and uses basic shapes, spatial reasoning, and manipulatives to model objects in the environment and to construct more complex shapes. The sample shows the naming of the basic shape and the model object, "a braslit," but the student has not described why the shape is a rhombus. To see if the student has developed understanding of Level 1, Analysis, the student would have to know that the shape is a rhombus because it is a four sided shape (a quadrilateral) with 2 pairs of parallel sides and four equal sides. This level of understanding is not expected of a Florida student until Grade 3.

By Fifth Grade, students are expected to move into Level 2, Deduction, where they identify the shapes within a class, but also can see the relationships between and among the properties.

(Click on the work to make it larger.)

Fifth Grade Student Work

Questions on the Student Sheet:

1. A square is a kind of rhombus. How can this be?

2. Name all the shapes above that are parallelograms. How can they be parallelograms and have other names as well?

3. An equilateral triangle is isosceles. How can this be?

4. Some obtuse triangles are scalene. Some obtuse triangles are isosceles. Sketch one or two examples of each.

5. Obtuse triangles cannot be equilateral. Explain why this is true.

This Fifth Grade assignment, Some Shapes Fit Many Categories, asks questions to assess students' geometric thought. Students who master this assignment are able to identify, draw, and analyze shapes, and are able to make connections about the relationships among their properties. They have moved into Level 2 of Geometric Thought. Students who have not yet mastered the assignment are likely at Level 1 of Geometric Thought, Analysis. After the teacher analyzes the student work he/she will be able to assess which students need more exploration with the concept through small group instruction.

When the Standard Snapshots were turned in, I was intrigued to see that both Kindergarten and Fifth Grade were sending home work in Geometry. It got me thinking about where students begin their geometry journey and where we expect them to be when they leave elementary school. It also got me thinking about the process that occurs for that learning, and the key role that each teacher plays with instructional delivery along the way. In addition, it made me go back into the standards to explore the vertical alignment from K-5. I came across a document that aligns the comparisons and is an easy to read resource for teachers. They are able, through this one document, to see what students learned previously and what students must be able to master during instruction in their classroom. It reaffirmed for me the importance of each piece of the puzzle in their journey, so students master the necessary skills and concepts before they emerge as middle schoolers.

Standard Snapshots of Student Work

At Chets Creek, our central focus has always been on students' academic performance. The work students produce lets us know whether our targeted instruction has worked or whether we need to reassess our path. When you stroll through our hallways, you see student work aligned with standards on each teacher's bulletin board. We post the work so our colleagues can compare work across grade levels or content areas, parents can see work that meets the standard, and students are exposed to the level of work they are expected to produce. The student work makes our classroom instruction more transparent and visible.

In addition, at weekly Teacher Meetings, it is not uncommon for teachers to bring student work to analyze. Their collegial conversation about the instruction that led to the students' products often has them exchanging valuable instructional ideas.

Years ago, in an effort to educate all of our parents, even those that are unable to stroll through our halls, we decided to package our standards-based bulletin board in a handout to be sent home with each student. The Standard Snapshot, as we called it, would be produced by each grade level and content area to go home with our students. In addition, to the selected grade level piece that is copied on the back of the handout, we also staple their own child's work on the assignment.

To prepare for a Standard Snapshot, each grade level / content area team meets to discuss work that is currently being produced. They agree upon a common assignment and they collect all students' work from the task. The teachers meet to review, compare, and select the student sample. They select work that meets the standard rather than work that exceeds the standard, because we want to educate parents about what their child's work is expected to look like. After the Snapshots are written, teachers turn them in to me with the original piece of student work; I edit / revise them, and turn them in for copies. After copies are made, the teacher attaches each child's work to the Standard Snapshot, and sends them home with students.

Creating a Standard Snapshot has been a parent communication piece for the past eight years, and is a powerful venue for collegial dialogue about student work among our teachers. It is a collegial practice that focuses on students' work products as they relate to the standards and is a practice I continue to find valuable and informative.

Second Grade Science Standard Snapshot

(To enlarge, click on the picture.)

Second Grade Student Work Sample Printed on the Back of the Snapshot

My Child's Work That Was Attached to the Snapshot

Grade 2 Snapshot

I took a focus walk of Second Grade on Thursday. I wasn't focused on any one thing (sometimes I am), rather this snapshot was simply to see how smoothly students and teachers had settled into a new year.

My first impression was a good one. They have settled in quite well, rituals and routines were getting firmly established, and classroom instruction had begun. The unannounced snapshot gave me a glimpse across content areas with lessons in technology, Readers' Workshop, Writers' Workshop, Math Workshop, and Science Workshop.

I visited a co-teach classroom where students were watching a video clip from '>You Tube which sang all of the countries in the world. Next, the teachers displayed their classroom blog on the board, read the blog with the students, and then read the comments. They modeled for students how to add a comment to the blog. Students were soaking in their every word. These students will now have a line of interactive communication with their teachers even when they are not in the classroom.

Next, I visited the English Language Arts rooms. (We are departmentalized from 2nd-5th grade so some teachers teach ELA and some Math/Science/Social Studies.) In one room, I found students actively involved in the active involvement part of their mini-lesson. Students were practicing the habits of a good listener. The teacher was facilitating instruction by listening in on the partners. This lesson is one of our districts 12 anchor lessons that serves as a foundation for setting up a successful Readers' Workshop structure for the year.

In two other rooms, I found students in the Work Period part of the Readers' Workshop. Students were in their book nooks with their Books in a Bag reading, were selecting new books from their leveled libraries, or were diligently adding a text to their book log that they had just finished. The teachers were monitoring the classroom to be sure that students were following the rituals and routines of the Work Period. Teachers need to make sure that students are building their independence and reading stamina. This is critically important, because on my next visit, during student work period, teachers will be administering DRA 2's to the students. After that, they will be using the DRA 2 data and diagnostic data to run guided reading and strategy groups. In order for the teacher to be doing the student conferring or small group lesson, the other students in the classroom will have to be actively reading and not interrupt the teacher.

Writers' Workshop was in full swing in another classroom as students diligently created stories in their sourcebooks while the teacher conferred with students. Evident to me was the students' confidence and writing fluency. Kudos to the Kindergarten and First Grade teachers who had students writing everyday and who held high expectations for the writing product. The 2nd Grade teachers have picked up where you left off. :)

In several other rooms, Math Workshop was off to a great start. Students were working on creating Combinations of 10 books as teachers encouraged and conferred with students. It was evident that every math teacher was on the same page and students in each class were working toward the same standards. Some students were working on putting a string of three numbers together to make 10 (5+3+2) while others were working on putting five numbers together to make 10 (1+1+2+3+3). These number strings will be used to construct a complete Combinations of 10 book. I can't wait until my next visit to see the completed book and to ask students what they learned about number strings.

The last classroom stop was into a Science class. The students were learning characteristics of specific animals including the plants they ate. The students received challenging riddles that they were charged to solve by matching them with the correct animal. This activity not only focused on science but also reading. The riddles had to be read and understood before the matching could take place.

In summary, I not only noticed that things were well underway in 2nd Grade, but also that this group of teachers has bonded as a team. Outside each of their classrooms is a sign hanging with their name, their hallways are prepared for student work, the artifacts in their rooms are similar, and their instruction is well aligned.

Standard Snapshot

Like other schools, we communicate with parents in typical ways through parent/teacher conferences, PTA nights, phone calls, e-mails etc... In addition, we communicate with parents via a weekly newsletter, The Connection, sent home from the Principal each Friday, a classroom newsletter sent home from each classroom teacher on Mondays, and through classroom blogs. We also use a unique tool we created about six years ago, The Standard Snapshot.

The Standard Snapshot is a Standards-based Bulletin Board in a parent package, and is sent home monthly beginning in December when grade level work begins to meet the standard. The snapshot is color-coded by grade level so it is easily identifiable to parents. The top of the snapshot reads... "This is a "snapshot" of a second grade lesson. We are sharing this with you so you may gain a better understanding of the work that your child is expected to do to meet the grade level standard. Attached to this standard piece of work is a sample of your child's work."

The front page of the Standard Snapshot includes:

What is the standard?

Why teach this lesson?

Tell me about this lesson.

Why does this student work meet the standard?

The back of the Standard Snapshot is one student's sample piece of work selected from the grade level. This piece of student work meets the standard. (We do not use work that exceeds the standard.) Then, their child's piece of work from the same assignment is stapled to the Standard Snapshot. The idea is for the parent to be able to compare their child's work to a standard piece of work.