Facilitating Developmental Learning

There are a number of ways to support developmental learning in the classroom; in the context of IMAGES we suggest creating a combined approach based on the menu of ideas listed in this section. Only if teachers are constantly reevaluating and readjusting their own strategies to teaching will students continue to learn in new and developmental ways.

Use Concrete Representations
Provide Time to Play
Use Examples and Non-Examples
Introduce and Implement Technology
Use Context and Prior Knowledge
Actively Engage Students
Implement the Equity Principle

Use Concrete Representations

Concepts such as symmetry, translations, reflections, area, and volume can be very abstract and are often not connected to the student's world. When a new concept is introduced, students need tactile and visual experiences to assist them in understanding the concept.9 For example, area and volume become more real when students cover a rectangle with squares or fill a rectangular solid with cubes. Using manipulatives allows the students to relate new ideas to things they already know and understand. After students feel comfortable with the concepts using manipulatives, students are ready to move from the concrete to the visual. For example, a figure drawn on paper could be folded along its line of symmetry to verify that it is in fact symmetric. The best use of instructional time is to wait to introduce some abstraction until students have moved into van Hiele's Level 2 (analysis).

A suggestion for instruction using concrete representations

Use a mirror or Mira™ to illustrate that a pattern block is symmetric about a line. The equilateral triangle has three lines of symmetry and the square has four lines of symmetry. Ask the students to find them.
Have students trace a flat plastic bear (or other animal) in one position, and then move or translate the bear to a new position and trace it again. Explain that this is an example of a geometric transformation, in particular a translation. Ask them: "Are the new figure and the original figure congruent?"
Have students place the bear on one side of a line drawn on a new sheet of paper. Then place a Mira™ on the line. Have them draw the bear they see reflected in the Mira™ on the other side of the line. Students now have a reflection of the bear across the line of reflection.
Ask them to look at properties of the reflection and compare these properties to what happens when they look at their own reflection, or wave, or wink in a mirror. For example, they should note that a particular point on the bear and the reflection of this point (its image) can be connected with a line segment, and that the line of reflection is perpendicular to this line segment and bisects the line segment.

Provide Time to Play

Students can learn significant mathematics-particularly geometry-during play.10 For example, when playing with tangrams students discover that they must sometimes "flip" a piece over to make it fit into a picture they are creating, while other times they must rotate the piece. Translating a piece from one location to another can change the picture they are creating from one form to another. Students discover that several pieces can be combined to make a larger piece and that the square can be broken up into the seven tangram pieces.

Also, if students are to use a manipulative to discover a mathematical concept, they first need time to play with the manipulative to become familiar with it. The same is true with computer programs, computer games, and calculators. Once familiar with the new tool, they will stay on task longer as teachers introduce guided activities.

For an example of how to incorporate time to play, see the Pattern Block Shapes activity.

Use Examples and Non-Examples

Students are likely to understand concepts, such as mathematical definitions, if a teacher provides them examples of a concept and non-examples.11 For example, students who are still at van Hiele's Level 1 (visualization) will not develop a solid understanding of what it means to be a triangle if they only see equilateral triangles oriented in the same way. To move students to the next level, a teacher needs to show them examples of similar objects that are not triangles, as well as obtuse and scalene triangles and triangles where the base is not horizontal. Using this approach when introducing each new figure will help students later in their schooling to place figures appropriately in the hierarchy of figures.12

Examples of a triangle		Non-examples of a triangle

Introduce and Implement Technology

Technology in the classroom is an effective way to engage students, while also supporting the development of skills that will be critical to them in future years. Appropriate uses of technology can also foster students' understanding of concepts in geometry and measurement. Technology can also assist the teacher in supporting students with special needs. On the other hand, use of inappropriate technology or ill-planned use of the technology can be worse for students than using no technology. A computer program that is too sophisticated for the students will only frustrate them. Using the computer for nothing more than drill and practice does not take advantage of technology's strengths.

Software such as Logo (turtle graphics) and MicroWorlds™ allows teachers to create activities and pose problems that encourage students to learn mathematics in the context of a problem and help children visualize geometric concepts and properties.13 For example, students can use geometric concepts to create a house or other shape. Simulations allow students to explore topics they may not be able or willing to do on their own. Students may be able to explore with virtual manipulatives in a way that would not otherwise be possible. Software such as the Geometer's Sketchpad® allows students to explore shapes like triangles and quadrilaterals in a dynamic environment.14 Students can also use the software to transform a figure, helping students to understand geometric transformations. Using the software to measure lengths and areas allows students to explore the concepts of area and perimeter. In addition, using calculators can allow students to focus on conceptual understanding and mathematical reasoning rather than solely on computation.

Using the Geometer's Sketchpad® to Explore Area and Perimeter

Create a triangle using Sketchpad:
1. Plot three points.
2. Select the three points.
3. Have Sketchpad create the polygon interior-a triangle.
Find the perimeter and area of the triangle:
1. While still selecting the interior, have Sketchpad measure the perimeter and area of the triangle.
2. Grab one of the vertices and move it around the screen until you have a triangle that is close to an equilateral triangle. Record the area and the perimeter.
Dynamically alter the triangle and observe what happens to the area and perimeter:
1. Move the vertex to create a skinny obtuse triangle (a triangle with one angle with measure more than 90 degrees) that has approximately the same perimeter. What has happened to the area?
2. Can you make the perimeter larger than the perimeter of the equilateral triangle while keeping the area smaller than that of the equilateral triangle?
3. Which triangles have the smallest area if you keep the perimeter (approximately) the same?
4. If you keep the triangle an equilateral triangle, what happens to the area when you increase the perimeter?
5. If you try different triangles with (approximately) the same perimeter, which triangle has the largest area?
6. Create different triangles with approximately the same area as the recorded area of the equilateral triangle. Can you find any with a much larger perimeter?
Repeat this investigation with quadrilaterals, starting with a square.
Repeat this investigation with pentagons, starting with a regular pentagon (all five sides are equal in length and all five interior angles have the same measure).

Students should discover that, if the new polygon is similar to the original polygon, increasing the perimeter increases the area.

Example 1

If the figures are not similar, increasing the perimeter will often, but not necessarily, increase the area.

Example 2

It is possible to have many different areas with the same perimeter.

Example 3

For a certain perimeter, the largest area occurs if you use a regular triangle (the equilateral triangle) for triangles, a regular quadrilateral (the square) for quadrilaterals, a regular pentagon for pentagons, etc.

For a certain area, the smallest perimeter for a particular type of polygon will occur if the polygon is a regular polygon.

Example 4

Use Context and Prior Knowledge

Students try to use what they already know to make sense of new mathematical concepts.15 Good teachers make these connections explicit, determining what their students already know and helping students see how the new concept is related to that knowledge.

For example, students may understand area as the number of squares required to cover a region or figure and understand perimeter as the length of string required to surround the region. A teacher can approach the concept of volume as a natural extension of the concept of area; it is the number of cubes required to fill an object or open figure such as an open box. Similarly, surface area is the three-dimensional equivalent of the two-dimensional concept of perimeter; it is the amount of material needed to surround the object. Students will remember and have a stronger conceptual understanding because the teacher built on prior knowledge.

A teacher can make this connection even stronger by using objects that students are familiar with (such as a cereal box) and by introducing the concepts as part of a problem ("How much cereal will fit in the box and how much paper is needed to cover the cardboard?"). Because it is difficult to transfer knowledge from one discipline to another, students should have experience applying mathematical knowledge to other disciplines. This will also motivate the need for learning mathematics.

Integrating mathematics and science or mathematics and art instruction is an excellent way to facilitate teaching in context. Small group work that asks students to investigate a concept or explore possible solutions to a problem encourages them to discuss what they already know and communicate mathematically how this knowledge could be applied in a new situation. Clustering ideas, as discussed in the “Teaching Strategies” section, is another way of helping students see connections within mathematics.

Actively Engage Students

"Students' understanding will increase if they are actively engaged in tasks and experiences designed to deepen and connect their knowledge of mathematical concepts."16

An ideal atmosphere for actively engaging students is one in which:

student discussion and collaboration are encouraged
students are expected to justify their thinking
students feel comfortable questioning the reasoning of the teacher and other students
students respect each other and the teacher
students feel comfortable making conjectures and suggesting alternative problem-solving strategies
students feel comfortable presenting results.

This atmosphere requires the teacher to be a facilitator of learning rather than a dispenser of knowledge.

Teaching by Engaging vs. Teaching by Telling
Teaching by Telling		The teacher states the definition of a convex polygon and draws a picture of one (or several) on the board.
Teaching by Engaging		The teacher gives examples (verbal and visual) of convex and non-convex polygons and asks the students to create a definition (in writing, either individually or in teams) and to justify their choices. Students then share their definitions and try to derive one definition.

Implement the Equity Principle17

A critical aspect of supporting students in learning developmentally is providing each student an equal opportunity to learn-a goal that has been made explicit in the NCTM's Principles and Standards for School Mathematics. A teacher should provide equal opportunity for all students, regardless of gender or other personal characteristics, to learn mathematics; educational equity requires that teachers have high expectations for all students.18 Teachers are responsible for challenging their own biases, which may be unintentional. The information in this section is intended as a reminder of the goals that we as educators must constantly keep in mind.

Gender equity

The under-representation of women in mathematics-related fields has been well documented, and educational research has examined factors related to this and to the lower achievement of females than males in mathematics.19 Although gender differences in mathematics achievement have continued to decrease, teachers need to create learning environments where both girls and boys are equally encouraged to learn and achieve in mathematics. Approaches that bridge the gap between the sexes in learning mathematics include:

using cooperative learning
using cooperative rather than competitive mathematical games
calling on girls as often as boys
allowing sufficient wait time for a response to a question
praising both sexes for their responses
creating an atmosphere that encourages all students to take risks
having high expectations for all students
encouraging communication about mathematics
emphasizing the social nature of learning mathematics

Multicultural learning environments20

There is a pervasive belief in the United States that some students do not have the "math gene" and therefore cannot learn significant mathematics. Traditionally, these students have included students living in poverty, students for whom English is a second language, and non-whites. Research suggests that all students can learn significant mathematics if they receive a quality educational experience with the extra support that they need,21 but teachers must have high expectations for all students.

One way for teachers to combat the belief that some students cannot learn mathematics is to highlight the contributions to mathematics of different countries, including Africa, India, China, and South America. This awareness and the ability to identify with the mathematics accomplishments of others from similar cultural backgrounds can motivate some students to learn the mathematics teachers are presenting. For example, tessellations, symmetry, and geometric transformations are found in Native American and African designs.22

For research in this area, see James Hiebert and Thomas P. Carpenter, "Learning and Teaching with Understanding," in Handbook of Research on Mathematics Teaching and Learning, ed. Douglas A. Grouws (New York: Macmillan, 1992), pp. 65-97.
Van Hiele, "Developing Geometric Thinking," pp. 310-16.
Annie Selden and John Selden, "The Role of Examples in Learning Mathematics," The Mathematics Association of America Research Sampler 5 (February 20, 1998).
National Council of Teachers of Mathematics, Principles and Standards for School Mathematics, p. 122.
Michael T. Battista and Douglas H. Clements, "Using Spatial Imagery in Geometric Reasoning," Arithmetic Teacher 39, no. 3 (November 1991): pp. 18-21.
Azita Manouchehri, Mary C. Enderson, and Lyle A. Pagnucco, "Exploring Geometry with Technology," Mathematics Teaching in the Middle School 3, no. 6 (January 1998): pp. 436-42.
Hiebert and Carpenter, "Learning and Teaching with Understanding," p. 80.
John Sutton and Alice Krueger, eds., EDThoughts: What We Know about Mathematics Teaching and Learning (Aurora, Co.: Mid-Continent Research for Education and Learning [MCREL], 2002), p. 91.
For an extensive listing of equity and gender and multicultural education resources, see Andria P Troutman and Betty K. Lichtenberg, Mathematics a Good Beginning (Belmont, Ca.: Wadsworth/ Thomson Learning, 2003), pp. 570-72.
National Council of Teachers of Mathematics, Principles and Standards for School Mathematics, p. 12-14.
Gila Hanna, "Reaching Gender Equity in Mathematics Education," The Educational Forum 67, no. 3 (spring 2003): pp. 204-14.
For discussion of this topic and listing of resources, see Geneva Gay, "The Importance of Multicultural Education," Educational Leadership 61, no. 4 (December 2003/January 2004): pp. 30-35; Lynda R. Wiest, "Multicultural Mathematics Instruction: Approaches and Resources," Teaching Children Mathematics 9, no. 1 (September 2002): pp. 49-55; and Claudia Zaslavsky, "Exploring World Cultures in Math Class," Educational Leadership 60, no. 2 (October 2002): pp. 66-69.
Janet Trentacosta, ed., Multicultural and Gender Equity in the Mathematics Classroom: The Gift of Diversity (Reston, Va.: NCTM, 1997).
Lyn Taylor et al., "American Indians, Mathematical Attitudes, and the Standards," Arithmetic Teacher 38, no. 6 (February 1991): pp. 14-21; Claudia Zaslavsky, "Multicultural Mathematics Education for the Middle Grades," Arithmetic Teacher 38, no. 6 (February 1991): pp. 8-13. Other sources of multicultural mathematics activities include: Clare V. Bell, "Learning Geometric Concepts through Ceramic Tile Design," Mathematics Teaching in the Middle School 9, no. 3 (November 2003): pp. 134-40; Marina C. Krause, Multicultural Mathematics Materials, 2nd ed. (Reston, Va.: NCTM, 2000); Maureen D. Neumann, "The Mathematics of Native American Star Quilts," Mathematics Teaching in the Middle School 9, no. 4 (December 2003): pp. 230-36; and Lynda S. Paznokas, "Teaching Mathematics through Cultural Quilting," Teaching Children Mathematics 9, no. 4 (December 2003): pp. 250-56.