**while teaching a math methods class to future teachers. It was one of the first research models that I knew would work.**

*Conceptual Development Model*The

*C***involves three stages of learning: 1) concrete or manipulative, 2) pictorial, and 3) the abstract. The concrete stage involves using hands-on teaching which might involve the use of math manipulatives or real items. Next, the pictorial stage utilizes pictures to represent the real objects or manipulatives. A visual such as a graphic organizer would also fit in this stage. Last, the abstract stage of development entails reading the textbook, using numbers to compute, solving formulas, etc. Let's look at two classroom examples.**

*onceptual Development Model*

**Example #1:****You are a first grade teacher who is doing an apple unit. You decide to have the children graph the apples, sorting them by color.**

__Concrete__**:**Using a floor graph, the children use real apples to make the graph.

__Pictorial__**:**The children have pictures of apples that they color and then put on the floor graph.

__Abstract__**:**The children have colored circles which represent the apples.

*Example #2:*You are a fifth grade teacher who wants to teach how to find the volume**of a cube or rectangular solid.**

**Concrete****:**Bring a large box into the classroom, a box large enough for the children to climb inside, OR have the students build 3-D objects using multi-link cubes.

**Give the students pictures of 3D objects which are drawn but shows the cubes used to make the solid. Have the students count the cubes to determine the volume.**

__Pictorial__:

**Abstract****:**Have students use the formula l x w x h to find volume.

Requiring my perspective teachers to think about this model and to use it when planning a math unit dramatically changed the quality of instruction which I observed in the classroom.

Writing Lesson Plans |

*"Open your books to page...."*without any regard to the other two stages of development. Using manipulatives and graphic organizers have changed my students' ability to learn math, and some have even ended the semester by saying,

*"I like math"*. Maybe this is a model we should all consider implementing.

If you want more examples and suggestions about using this model to write lesson plans, click on the link below the illustration. It will take you to

**Graphing without Paper or Pencil**in which is appropriate for grades K-5 and is based on the Conceptual Model of Development: concrete to pictorial to abstract.