This learning experience is a modification of a Masters' Project I developed in Spring 2004. I was, and still am, teaching sixth-grade in an urban parochial school. This is now my sixth year teaching math. Nine years ago the school adopted a commercial math curriculum called Saxon Math: An Incremental Development. As the title suggests, the curriculum is spiral in nature. Instead of "chunking" the concepts (i.e. teaching and practicing one thing for a week and then moving on to the next idea), "This book consists of a series of daily lessons and investigations that are carefully sequenced to incrementally develop a spectrum of skills and concepts" (Hake and Saxon, p.xi). It presents a distinct concept and then inserts it into daily practice with a variety of other concepts. The rationale behind this format is that the student should build up a repertoire of skills that, when practiced consistently, will become automatic. The curriculum is strictly regimented, meaning that the authors of the program expect teachers to follow a specific daily routine in order to optimize results. As a novice teacher I found this "cookie-cutter" method of teaching to be very convenient. As I gain experience, however, I find the curriculum to be somewhat restrictive.
Prior to developing the Masters' Project, I noticed that many of my math students, even the ones who usually excelled at math, were struggling with certain math concepts. The Saxon program, while strong in many respects, relies heavily on pictorial representations and paper-and-pencil tasks to present topics. It seemed obvious to me that if these two modes of communication were inadequate for my students to master the information, I needed to provide them with concrete models that they could manipulate. I asked the school secretary for a copy of the Saxon program's ancillary materials catalogue so I could order the necessary items. She obliged, but I was disappointed to find that, at the sixth-grade level, the only materials available were posters (more pictorial representations).
While it is certainly easy enough to order materials intended for primary grades and adapt them to my own use, an important question nags at me. Why does this publisher (and others I subsequently found) think that math manipulatives are not appropriate for the middle grades? Manipulatives are necessary at the primary level for practical reasons; young children are not experienced with text and other modes of learning need to be employed to compensate. By the end of fourth grade children should be sufficiently skilled with text to deal written directions, explanations, word problems, etc., but that does not mean hands-on materials become obsolete. They should be employed in addition to the visual and verbal/abstract representations to enhance children's understanding.
Teacher educators Sheffield and Cruikshank (1995) point out that young children have a natural tendency to interact with objects in order to make sense of their world. As they begin formal schooling, this tendency should be fostered.
Children should be physically involved in mathematics….
Children use [hands-on] materials for counting, developing
patterns, creating, observing, constructing, discussing, and
comparing. From the manipulations and observations come
the abstractions of quantitative ideas and the communication
of these ideas in pictorial and, later, in symbolic form. (p.20)
The authors go on to state that the mathematical language that represents the concept cannot be understood prior to the physical experience. In other words, rote memorization and teaching through drill and practice are not effective if the child cannot form a connection between what is being taught and one's own prior knowledge. Without a mental referent, the child cannot comprehend the explanation of an unknown; the teacher may as well be speaking a foreign language.
Through the use of geoboards, I am trying to provide the mental referent my students need to comprehend the geometric formulas they so often confuse and misuse. I would like to thank Jeff Arnold for providing me with a class supply of 10 x 10 geoboards, which will be useful for working with the dimensions more often encountered by students in the upper grades.
I would also like to thank my classmates for making suggestions to improve my learning experience. The peer reviews underscored a need to incorporate more technology into the classroom. Carol Mullen recommended illuminations.nctm.org as a good website to use for enrichment activities. Teachers can search the site by concept or by grade level. Mike Guarnieri mentioned the use of the CAD program to move beyond the paper and pencil tasks. CAD stands for "computer-aided design" and is a drafting program that students can use to draw up plans for two-dimensional or three-dimensional objects.
During the peer reviews Pat Loncto and Lee Lyle noted a lack of real-world connection in my anticipatory set. Ways to address this could include constructing a quilt from individual squares, measuring floor space to select an appropriate area rug, and inviting professionals who use measurement in their work (ex. architects, carpenters, realtors) for a career day.
I was prompted by the peer review to devise a closure activity that provides a real-world connection. Students often complain about issues of space: "Stay out of my space, I need more space". Showing students how to identify the amount of space taken up by simple and complex shapes can transfer to measuring and manipulating their personal spaces. For example, they can use measuring tapes to measure the floor space in their bedrooms and the furniture it contains. Students can then record their "blueprints" on graph paper and experiment with alternative ways to make the most effective use of their spaces. This activity is empowering to middle grade students, who are growing physically and are craving more personal choice.
When writing this learning experience, my focus question was, "Does my learning experience follow the congruency table?" The peer reviews in November 2006 and May 2007 affirmed that I did indeed write a learning experience that met the following objectives for my students:
- Construct squares of various sizes and compile data to develop a formula for area of a square.
- Review the properties of rectangles and develop a formula for area of a rectangle.
- Compare the properties of right triangles to their corresponding rectangles and develop a formula for area of a triangle.
These objectives relate directly to the performance indicator: Determine the area of triangles and quadrilaterals (squares, rectangles, rhombi, and trapezoids) and develop formulas.
I attribute the high rate of my students' success to the direct correlation between the performance indicator and the learning objectives. I also think focusing on one performance indicator makes the experience easy for the teacher to manage.
During this learning experience many students realized the advantage of learning a geometric formula. In her exit visa from Day Two, Jordan wrote, "One formula helps you find area of all rectangles." Christian wrote, "I learned that A = l × w works for all rectangles." I believe these students will internalize this knowledge and retain it longer than someone who was simply told to memorize the formula.
The lessons using geoboards also prompted curiosity and discovery of Math concepts outside the stated performance indicator. After exploring the geoboards on Day One, Alida wrote, "Has anyone in your class ever made a circle in Math class on your geoboards? What does x and y axis stand for? Who created the geoboards?" All of these questions show an interest in Math that Alida can explore independently or we can explore as a class, increasing her engagement even further. Daryl observed, "Today I learned that using geoboards and rubber bands is a good way to find square root." Although the teacher's objective was related to geometry, Daryl was able to see a connection to number theory through the visual/spatial experience.
