Teaching for understanding with technology
HGSE Professor Stone Wiske
How can technology enrich learning and teaching experiences? In her book, Teaching for Understanding with Technology (Jossey Bass, 2005), Stone Wiske features the work of educators who use technology to make learning difficult topics more engaging and generative for students. An example is "Quilt Math," developed by the book's co-author Kristi Rennebohm Franz, which integrates technology into math lessons in an elementary school classroom.
One of the most important decisions that educators make is choosing topics worthy of time and effort in the classroom. The Teaching for Understanding (TfU) framework emphasizes generative topics—topics that are central to a given discipline; authentic, accessible, and interesting to students; fascinating and compelling for teachers; accessible through multiple "entry points" (i.e., ways of learning); and that generate and reward continued inquiry.
Kristi Rennebohm Franz, an experienced TfU practitioner, used technology to make math more generative and accessible in her elementary school classroom. She created "Math Quilts" to help first and second grade students understand a challenging topic—how to use math for analyzing and predicting patterns in the world.
On the first day of school, Kristi posted a 6'' by 6'' square of fabric on the classroom whiteboard. Asking the 6-8 year old students what they saw, students observed:
The patch has ladybugs on it.
The ladybugs are red.
Kristi encouraged students to make observations that involve math:
There is one square patch with nine ladybugs on it.
Adding one square of fabric each day, Kristi completed one row of the quilt by the fifth day of school.
Students noticed the pattern of background colors:
Yellow, Red, Red, Yellow, Red
Each morning, students were eager to see what the next patch would look like, and to document their observations. They began each day by writing individual comments on math ideas they observed in the quilt, to prepare for a whole class discussion. As students presented their comments to the class, Kristi wrote them on the white board and two students were assigned to enter those comments into a computer document.
Kristi used technology to document the students' Quilt Math learning in multiple ways. A digital camera allowed students to review the quilt's appearance over time. Daily class notes were combined with the digital photos into a Quilt Math photo journal, posted to their class website and shared with family members.
On school day #25, the Ladybug Quilt was complete.
Students continued to document observations about the patches in their electronic Math Journal. As you read some of the students' notes, it's easy to imagine how Kristi could use the Math Quilt to generate lessons on geometry, permutations and combinations, proportions, and other applications of mathematics:
We have a color pattern by rows:
1st row is [yellow,red,red,yellow,red]
2nd row is [red,yellow,red,red,yellow]
3rd row is [yellow, red, yellow, red, red]
4th row is [red,red,red,yellow,red]
5th row is [red, yellow, red, red, yellow]
We have 16 red backgrounds and 9 yellow backgrounds.
We have a pattern of fabrics by rows that goes [using letters to represent each fabric:]
We have a diagonal of five Ladybugs that goes from the top left corner to the bottom.
We have 5 Ladybug patches 6 large white dot patches, 5 tan dot patches, 4 red dot patches, and 5 small white dot patches.
Kristi designed a sequence of quilts throughout the school year to build the breadth and depth of students' understanding of math concepts. For instance, students observed geometric shapes embedded in their "Winter Quilt." They defined "pinwheels" as groups of four patches lined up such that all the points of their eight triangles met in the middle. The students identified three pinwheels in the quilt – indicated here using colored squares:
This discussion provided the opportunity to compare discrete or exclusive sets of triangles (i.e., the first and second row pinwheel, outlined in pink) with overlapping or nested sets (i.e., the pinwheels at the end of the 2nd and 3rd row, outlined in red, and in the middle of the 3rd and 4th row, outlined in green).
Technology enhanced this experience by giving individual students the opportunity to create and manipulate electronic quilts. Students used a software program called Shape-Up, by Sunburst, to create their own virtual quilts—and to meet learning goals in geometry.
With the software, students used color tools to identify and explain the multiple sizes of squares they created in the rows and columns, their composition (number of triangles making a square, number of triangles making a parallelogram, etc.) and relationships (e.g., nested and overlapping sets of triangles). Students printed out their designs as visual documents to accompany their writing about the quilt geometry. In whole-class and small group lessons and in parent-teacher-student conferences, students presented and discussed their original designs. (Visit Kristi's Winter Quilt Web site to follow the students' developing understanding of geometry.)
Starting from a 6'' by 6'' piece of fabric and culminating in an online photo journal, electronic work products, and parent-teacher-student conferences, Kristi Rennebohm Franz used technology to facilitate her students' understanding of difficult math concepts. As Stone Wiske's book explains, the Quilt Math curriculum represents Teaching for Understanding in action: providing multiple entry points into the topic, using visual, verbal, and symbolic methods; allowing the students to demonstrate their understanding; and enabling the teacher to assess learning.
Quilt Math materials created by Kristi Rennebohm Franz, and used by permission.