Math Teaching Method
Teaching Small Units
- Teaching Small units of information pertinent to a mathematical concept makes it easier for the students to learn.
- A teaching method that uses small, easily comprehensible units of information is much effective (Ausubel, 1969; Brophy & Everston, 1976).
- Using incremental steps when teaching new information helps students to grasp the concept easily (Rosenshine and Stevens, 1986 and Brophy and Everston, 1976).
Class and Home Practice
- Class practice and home practice enable students to retain the knowledge gained during class time and translate it into an automatic skill.
- Research studies have shown that students who are taught with a mathematics curriculum that uses continual practice and review demonstrate greater math achievement and skill (Good & Grouws, 1979; Hardesty, 1986; MacDonald, 1984; Mayfield & Chase, 2002; Ornstein, 1990; Usnick, 1991).
Reviewing Concepts and Methods Repeatedly
- Reviewing previously introduced math concepts and connecting those concepts with newer concepts help to construct a knowledge chain that helps the students to gain advanced math problem-solving skill.
- The benefits of review have been validated by research since the early part of the 20th century, and many studies suggest that when a review is incorporated into the learning process, both the quantity and quality of what is learned is improved (Dempster, 1991).
Continual Practice Supplemented with Review
- Studies in cognitive science also support the continual practice, because it develops computational automaticity—it increases retrieval speed, reduces the time required for recognition, and decreases interference (Klapp, Boches, Trabert, & Logan, 1991; Pirolli & Anderson, 1985; Thorndike, 1921).
- Anderson’s (1983) Adaptive Concept of Thought (ACT) theory explains the development of expertise through three stages: cognitive, associative, and autonomous.
- In cognitive stage, learners rehearse and memorize facts related to a particular skill that assists them in solving a problem.
- In the associative stage, learners are able to detect errors and misunderstandings through consistent practice.
- In autonomous stage, the skill becomes automated reducing the amount of working memory needed to perform the skill and leading to expertise with that skill.
- Review and continual practice enable knowledge retention and skill development.
The Direct Instruction (DI) model is the most carefully developed and thoroughly tested program for teaching reading, math, writing, spelling, and thinking skills to children. Siegfried Engelmann and Wesley Becker developed DI at the University of Illinois in the 1960’s. It was further developed by Engelmann, Doug Carnine, Bonnie Grossen, Ed Kameenui, Jerry Silbert, and others at the University of Oregon. Two major rules underlie DI: (1) teach more in less time, and (2) control the details of the curriculum.
Direct instruction (DI) is a general term for the explicit teaching of a skill-set using lectures or demonstrations of the material to students. With DI, every student can achieve academically if they receive adequate instruction (Becker et al., 1973). The Direct Instructional Systems in Arithmetic and Reading (DISTAR) program for arithmetic is designed to teach the problem-solving operations to ensure that students know how an operation works and why they are using it. The DISTAR program for arithmetic is designed to teach the problem-solving operations to ensure that students know how an operation works and why they are using it. Arithmetic facts are then taught after they can use the operations. Then they learn “several fundamental laws or rule of mathematics”.
Direct teaching takes many forms, ranging from the typical chalk-and talk or PowerPoint lecture – where students are mainly passive recipients of information – through to highly structured but interactive classroom sessions (e.g., the Direct Instruction model of Engelmann & Carnine, 1982). Regardless of the type of direct teaching being used, the teacher or instructor requires a repertoire of skills and competencies that cover:
- planning the content and method of delivery (including appropriate use of audio-visual equipment and ICT)
- managing the available time efficiently
- presenting the content in an interesting and motivating way
- explaining and demonstrating clearly
- knowing when and how to explain key points in more detail
- using appropriate questioning to focus students’ attention, stimulate their thinking, and check for understanding
- dealing with questions raised by students
- evaluating students’ learning and participation
- giving feedback to students.
Anderson, J. (1983). The architecture of cognition. Cambridge, MA: Harvard University Press.
Ausubel, D. P. (1969). Readings in school learning. New York: Holt, Rinehart, and Winston.
Becker, W. C. & Engelmann, S. (1973). Project description and 1973 outcome data: Engelmann and Becker Follow Through Model. Journal.
Brophy, J., & Everston, C. (1976). Learning from teaching: A developmental perspective. Boston: Allyn and Bacon.
Dempster, F. (1991, April). Synthesis of research on reviews and tests. Educational Leadership, 48, 71–76.
Engelmann, S., & Carnine, D. (1982). Theory of instruction: principles and applications. New York: Irvington.
Rosenshine, B., & Stevens, R. (1986). Teaching functions. In M.C. Wittrock (Ed.), Handbook of research on teaching: Vol. 3. (pp. 376–391). New York: Macmillan.
Good, T. L., & Grouws, D. A. (1979). The Missouri mathematics effectiveness project. Journal of Educational Psychology, 71, 355–362.
Hardesty, B. (1986). Notes and asides. National Review, 37, 21–22.
Klapp, S. T., Boches, C. A., Trabert, M. L., & Logan, G. D. (1991). Automatizing alphabet arithmetic: II. Are there practice effects after automaticity is achieved? Journal of Experimental Psychology: Learning, Memory, and Cognition, 17, 196–209.
MacDonald, C. J. (1984). A comparison of three methods of utilizing homework in a precalculus college algebra course. Unpublished dissertation, Ohio State University, Columbus.
Mayfield, K. H., & Chase, P. N. (2002). The effects of cumulative practice on mathematics problem-solving. Journal of Applied Behavior Analysis, 35, 105–123.
Ornstein, A. C. (1990). Practice and drill: Implications for instruction. National Association of Secondary School Principals, 74, 112–117.
Pirolli, P. L., & Anderson, J. R. (1985). The role of practice in fact retrieval. Journal of Experimental Psychology: Learning, Memory, and Cognition, 11, 136–153.
Thorndike, E. L. (1921). The psychology of drill in arithmetic: The amount of practice.The Journal of Educational Psychology, 12, 183–194.
Usnick, V. F. (1991). It’s not drill and practice, it’s drill or practice. School Science and Mathematics, 91, 344–347.