# Concept Attainment in Math Education

Developing Conceptual Thinking: The Concept Attainment Model
Effective teaching in mathematics requires conceptual learning, inductive thinking, cooperative learning, critical thinking and dialogue in the classroom. The concept attainment model embodies all of these attributes. The following paper will summarize the theory and four phases of concept attainment as outlined in the research article “Developing Conceptual Thinking: The Concept Attainment Model”, (Johnson, Carlson, Kastl and Kastl, 1992). Then, it will illustrate how this strategy can be applied to the
K-12 mathematics classroom.

Here is an example for math class:

Barrie Bennet and Carol Rolheiser, Beyond Monet, The Artful Science of Instructional Intelligence, 2001. Toronto, Ontario Canada

The Concept Attainment Model (Bruner, Goodnow and Austin, 1967) provides the student with a collection of examples that represent the particular concept and a collection of examples that do not represent the concept. The students use Bloom’s analysis and synthesis thinking skills, as well as prior knowledge, to categorize the examples and create a conceptual definition for the concept. Once a concept definition is obtained, the students use their new conceptual learning to categorize the new examples and non-examples. The final stage involves creating new examples and non-examples that fit into the new concept’s definition. The following is a summary of the four phases as outlined in the research article.

This is a math resource that uses Concept Attainment for Arithmetic Sequences,

Phase One: The students are presented with a data set that contains ‘yes’ and ‘no’ examples. The data set can be a concrete collection of items that represent a concept, or it can be a list of yes and no examples. The examples and non-examples could be provided on index cards and the teacher can present them three to five at a time, or the list could be provided at one time to the student. Next, the student creates a hypothesis for the concept based on all of the examples and non-examples. Since cooperative learning improves student achievement (Leikin and Zaslavsky, 1999) an option would be to have the students work alone first, with a partner, then with the class in a Think-Pair-Share strategy. This would enable the ELL and LD student ample time to process the new information and would reduce the stress that can lead to math anxiety. Differentiated learning could be introduced if some groups are given more challenging data sets and some groups given more simplistic data sets. To encourage involvement of the ELL students, pair them with a peer they trust (Marks Krpan, 2007).

Phase Two: Students share their hypothesis with the class and the characteristics of the concept are summarized by the teacher. The students then identify the characteristics that separate the yes examples from the no examples. The teacher facilitates classroom discussion and encourages dialogue. Mathematic discourse in the classroom is critical in promoting math understanding (Stein, 2007). The essential and non-essential traits are also mentioned. For example in math, if the data set were to contain examples and non-examples for standard form of a line (see attached), the non-essential characteristic would be the variable type x, y or w, whereas the essential characteristic may be the order in which the variable is presented in the equation, i.e. variables must be stated in alphabetic order in an equation in standard form.

Phase Three: Students test their new conceptual learning by categorizing new examples into the yes or no category. Students must justify their reasoning. Students then use Bloom’s synthesis skills to create their own yes and no examples and justify why these examples and non-examples fit into the conceptual definition.

Phase Four: Students reflect upon their thinking. They reflect on the thinking patterns and processes that lead to the conceptual definition. Reflection of conceptual learning assists the students in solidifying their new knowledge and allows them to see the variety of procedures other students may have used to formulate the same conclusion yet in a different manner.

Last, the article outlines how the model could be used in other areas, but only outlines how it could be used in physics. The students could be given data sets that contain different physics activities or experiments that relate to a certain concept.

Considering the Concept Attainment strategy is a powerful teaching technique, research and published resources related to mathematics are very limited. Concept attainment data sets could be used effectively in the K-12 classroom. Concepts such as addition, subtraction, symmetry, etc. could be used.

In summary, concept attainment helps students make connections between what they already know what they will be learning, promotes students to go beyond merely associating a key term with a definition and most importantly improves the conceptual learning of mathematics for all learners.

Bruner, J., J.J. Goodnow, & G.A. Austin. 1967. A study of thinking. New York Science Editions.

Johnson, J., S. Carlson, J. Kastl, & R. Kastl. 1992. Developing conceptual thinking: The concept attainment model. The Clearing House, 66(2), 117-121.

Liekin, R., & Zasalavksky, O. (1999). Cooperative learning in mathematics. The Mathematics Teacher, 92(3), 240, 246.

Marks Krpan, C. (2007). Exploring effective teaching strategies for ELL students in mathematics classrooms. (C. Rolheiser, Ed.) School University Partnerships: Creative Connections, OISE.

Stein, C. (2007). Let’s talk: Promoting mathematical discourse in the classroom. The Mathmematics Teacher, 101(4), 285-289.

Developing Conceptual Thinking:
The concept attainment model.

A Summary

Johnson, J., S. Carlson, J. Kastl, & R. Kastl. 1992. The Clearing House, 66(2), 117-121.