Helping Preservice Teachers to Teach Mathematical ConceptsAuthor(s): Edward J. DavisSource: The Arithmetic Teacher, Vol. 31, No. 1 (September 1983), pp. 8-9Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41190739 .

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Teacher Education Helping Preservice Teachers to Teach

Mathematical Concepts By Edward J. Davis

University of Georgia, Athens, GA 30602

Teaching mathematical concepts is important. In a time when students' performance on basic skills is being emphasized, , teachers must be alert not to stress computation at the ex- pense of concepts. Understanding the meaning of a fraction differs from knowing how to add fractions. A lack of conceptual knowledge of a fraction can contribute to incorrect perform- ance in computational skills with frac- tions. The same can be said about other mathematical concepts, such as percent, decimals, division, factoring, like terms, square root, and so on.

What Are Mathematical Concepts? Simply put, a mathematical concept is a collection of meanings that one as- sociates with a word used in mathe- matics. The set of associations you have with the term altitude consti- tutes your concept of altitude. These associations or meaning did not devel- op all at once, and they will become more numerous as you learn more about altitudes.

Open any school mathematics text- book to the index and begin reading. You will be bombarded with mathe- matical concepts and the associations concerning these concepts that are introduced, reviewed, and extended throughout the text. The main head- ings of the index of the text that I have before me begin like this:

Addends Addition Algorithm

Area Attribute Pieces Center Centimeter

Six of these first seven entries are mathematical concepts.

We are teaching a mathematical concept whenever we teach the mean- ing of a word used in mathematics. The extent of a student's conceptual knowledge is probably a strong indi- cator of this student's performance in class, on tests, and in applying mathe- matics to solve problems.

How Can We Teach Mathematical Concepts to Children? I see at least six main tools, or moves, that mathematics teachers can use to help students create a well-balanced set of associations with any given word in mathematics. They are pre- sented in the following example that deals with the concept of percent. The teaching tools, or moves, are generat- ed from six types of questions that can give evidence of a student's knowl- edge of percent.

Question type Teaching tool (setting) (move)

1. Can you give Examples some examples of 30 percent, 60 percent, 100 percent, and 150 percent of these objects or collections?

2. Which of these Nonexamples pictures defi- nitely does not show 25 per- cent of the ob- ject or collec- tion? Why not?

3. Think about a What's True can of Coke. About what percentage of the can would someone drink if they took -

a sip? a gulp? half the can? almost the whole can? the whole can? two cans and a sip more?

4. Rex claims that Testing he has cut off 30 (Guarantee) percent of this string. How can we check to see?

5. What is the dif- Compare- ference be- Contrast tween 40 per- cent and 80 percent of this segment? How is 120 percent of a shoelace like 20 percent of the shoelace?

6. What do you Definition think the word percent means?

Keep in mind that each of the six settings listed on the left is just one possible way of helping students ac- quire the corresponding kind of asso- ciation or meaning labeled on the right. Students can be said to have an understanding of a concept to the ex- tent that they can successfully make or respond to moves involving Exam- ples, Nonexamples, What's True, Testing, Compare-Contrast, and Defi- nition that are made or posed by the teacher. These moves can be sponta- neous questions or statements or those made in response to students' questions. Posters or bulletin boards

8 Arithmetic Teacher

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frequently display situations involving Examples or Whaťs True. Tutorial sessions at a computer frequently draw from these six kinds of under- standing about mathematical con- cepts.

Training Preservice Teachers to Teach Concepts To the extent that course evaluations by student teachers are valid and that my observations of lessons taught by them are accurate, I have had success in training preservice teachers to teach mathematical concepts by using peer teaching and microteaching. The lessons are ten to twenty minutes in length. Preservice teachers are urged to proceed according to the following format whenever possible.

1. Set an objective for the lesson, motivate students, and review the ap- propriate knowledge.

2. Present four or five examples of the concept - make them significant examples. Try to use some concrete representations.

3. Present two or three nonexam- ples of the concept. Use some nonex- amples that are reasonably close to being examples and point out why they are not examples.

4. Attempt to get the students to identify what's true about the concept through the use of your examples and nonexamples (it is hoped that they are still in view).

5. Have students practice identify- ing and classifying situations as exam- ples or nonexamples of the concept.

6. Discuss students' selections and identify those tests that help students decide if an object is an example.

7. If feasible, compare this concept to a related concept known to the students or contrast different exam- ples of the concept to show similar- ities and differences.

8. Have students define the con- cept using their own terminology. Consider comparisons to the textbook definition.

9. Provide more practice for the students.

A few years ago, the thought of prescribing for teachers a sequence of steps for a lesson plan was disturbing to me. I felt that preservice teachers should use their own creativity and artistry in sequencing the lesson on concept development. Nonetheless, they seem more successful with this new format than with the nondirected approach. Let me add a few more observations:

• Students are to follow it three or four times and then begin to change it according to their beliefs and ex- periences.

• Research by Dossey (1976), Shum- way (1974), and others indicates no advantage in any particular se- quence for teaching mathematical concepts.

• If the lesson does not proceed as smoothly as students would like, they can share the "blame" with the instructor.

Prescribing a lesson format may keep preservice teachers from being overwhelmed by theory and give them some security. Student teachers delib- erately make changes in this format within the span of a few lessons and are sometimes "forced" to depart from it in response to students' ques- tions. These variations are good. The prescribed format, although appearing to control teaching behavior, still leaves the teacher considerable lati- tude in the choices for examples, non- examples, and physical referents used in the presentation and in the practice.

References

Davis, Edward J. "A Model for Understanding Understanding in Mathematics." Arithmetic Teacher 26 (September 1978): 13-17.

Dossey, John A. "The Role of Relative Effica- cy Studies in the Development of Mathemati- cal Concept Teaching Strategies: Some Find- ings and Some Directions." Teaching Strategies: Papers from a Research Work- shop. Columbus, Ohio: ERIC/SMEAC and Ohio State University, 1976. (ERIC Docu- ment Reproduction Service No. ED 123 132)

Shumway, Richard J. "Negative Instances in Mathematical Concept Acquisition: Transfer Effects between the Concepts of Commuta- tivity and Associativity." Journal for Re- search in Mathematics Education 5 (Novem- ber 1974): 197-211. m

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