## Discovery as a method of learning

The article of Marvin L.Bittinger “A Review of Discovery” discusses theoretical and practical aspects of discovery learning as well as discovery in mathematics. Undoubtedly, Bittinger has carried out deep research of literature and scientific sources to make such a review of discovery learning process. The author of the article analyzes a great number of works concerning discovery learning. Thus he succeeded to prove that discovery as process and result is an effective method of learning.

From the very beginning of the analyzed article, Marvin L.Bittinger introduces different definitions of the notion of discovery and presents his own view. He supports the ideas of the scientists who state that discovery as a method includes the following methods: the inductive, the nonverbal awareness, the incidental learning, the deductive and the various methods. Besides, Bittinger explains these methods in an accessible way giving illustrations of their use by the students in the learning process.

Concerning the structure of the article it can be considered as an effective one since it is composed of chapters that make the reading easier and more comprehensible.

Theoretical part is based on the Bruner’s assumption of the benefits of discovery learning which is increasing intellectual potency, intrinsic reward, learning of discovery heuristics and memory conservation. To explain the mentioned benefit, the author of the article uses bright examples of different scholars like Beberman, Suchman, Ausubel and others. At the same time, Bittinger provides the reader with contradicting points of view of different scholars giving possibility to choose what is more applicable to the audience. Along with the benefits of the discovery, drawbacks of its use are mentioned in the analyzed article. The method of discovery learning is a time-consuming process since it is not possible to wait eternally for discovery.

Practical part of the article deals with the results of the research in two research areas which are laboratory and classroom in nature. The author illustrates studies of various scientists and their experiments on groups of students. It is obvious that students trained to discover are more fluent than those taught in a traditional way. Moreover, a teaching inquiry or discovery is discussed on the ground of practical results of the researches carried out by different scholars. It makes the article trustworthy and interesting for the readers.

In the final part of the article Bittinger reviews the place of discovery in mathematics. The author highlights the number of writers in literature who also discuss the use and philosophy of discovery in teaching mathematics. Having analyzed different works concerning the discovery in mathematics, the author makes the conclusion that discovery should not be overdone.

## Theories of learning

Margaret E. Gredler’s article deals with the theories of learning and analyzes seven theories that have taken part in the formation of educational system and influenced the learning process. The analyzed text, being a part of the book “Theory into Practice”, includes lots of detailed information and brief explanations of concepts.

The author begins her article with the description of the origins of learning process highlighting the birth of experimental investigations in psychology. Then she discusses three theories that emerged from the experimental investigations which are Pavlov’s classical conditioning, Thorndike’s instrumental conditioning and the Gestalt perspective. In addition, she presents contemporary theories contributing specific variables to the learning process. Furthermore, the author presents other theories showing development of the learning process and demands to it up to the present system of education. However, each of the presented theories has developed a set of principles of the nature of learning and cognitive development.

Gredler’s article discusses various contemporary issues on universal education. She presents the broadest three issues which are of great importance for the future; they are the role of technology in learning, the learner’s concept of self and the influence of the culture of the learner.

The first important issue, the role of technology, is shown in the review of the works of Skinner and Gagne. In order to illustrate their ideas, Margaret E. Gredler gives example of using microcomputers by children. The author highlights the results of the experimental investigation and sums up that technology is to be restricted in early years. The reason for such conclusion is that there is no interpersonal interaction for children using technology. All in all, technology should be an assistant in the learning process, but not the deliverer of instructions.

The issue of the learner’s concept of self is discussed on the ground of the works of Bandura and Weiner. Self-efficacy and self-esteem are the notions introduced to support the concept and to illustrate the acquisition of skills in the process of learning. Along with the concept of self, there is an important relation to the previous concept of cultural effects. This is another important issue discussed in the article. The description of the influence of the culture is based on Skinner’s differentiation of cultures in time-space. According to the scientists the surrounding culture influences child’s mental functions. This idea is supported by examples and vivid illustrations which help the reader to believe the presented concept. Special attention is paid to Vygotsky’s researches devoted to cultural heritage and social interactions of children.

## Exploration of mathematical engagement

The article “A Phenomenological Exploration of Mathematical Engagement: Approaching an Old Metaphor anew” written by Handa Yuichi deals with the possibilities offered by mathematics for the person. The author describes an experience of learning mathematics that resulted in self-opening, deepening and lived experience. In order to study this process he sets forth the questions to be answered in the article and carries out experiment on five people.

Handa Yuichi discusses different notions which he presents in the form of threads to be followed by the readers and adherents. He gives comprehensive and clear explanations of the notions by means of illustrating situations taken from real life. Such notion as “frustration as deception” is explained in the example of the talk with a professional mathematician. In addition, he implements a notion of circle which is the base of his proofs and is the main symbol of the theory. Special attention is paid to “pushing over the edge” which the author supports by his own examples from his student’s life. It makes the analyzed article interesting and appealing to the audience.

“Frustration as disappointment” is the other notion introduced by Handa which is also explained on the ground of his own experience. Moreover, he uses the works of different scientists to prove his idea, for example, biologist T.H. Huxley.

It should be mentioned that the analyzed article is special because of the great number of images used by the author to show his view. Moreover, he uses metaphors for showing mathematical problems and in order to keep thinking of them. Along with the metaphors, Handa makes his article bright by means of many citations and dialogues. Such a form of explanation of difficult notions is perceived as an interesting story and deductive material at the same time.

Thus Handa Yuichi comes to the conclusion that mathematical engagement is meant in the engagement itself. In other words, it means that “becoming connected to the other one transcends the self”. Even these explanations have been taken by the author as a citation and after explained on the basis of his own experience and opinion. The article which is analyzed requires great attention due to its rich presentation of the material and big amount of sources used by the author. The broad use of metaphors and other literary means in the scientific text on mathematics makes this article interesting and special at the same time.

## Transfer of training

The article written by Gertrude Hendrix deals with educational psychology and the problem of using acquired knowledge in practice. In order to make research of the mentioned problem, the author illustrates experiments carried out in a psychology-of-learning course at the universities of Chicago and Illinois. The results of the experiments are presented in the form of hypotheses. For the purpose of clear understanding of the experiments carried out at universities and hypothesis stated by the author, Hendrix describes three methods that have been the base of her research. Moreover, the hypotheses stated in the article are studied carefully for revealing contradictions.

Along with the described experiment, Gertrude Hendrix makes a review of the works of different psychologies who were dealing with verbalizing of generalization and transfer power. Such theories as Gestaltist and conditions are to be modified in author’s opinion if the hypothesis is verified. Moreover, she gives examples of different training programs used in the schools and criticizes them. Two postulates from Dewey’s *Experience and Education* are considered inconsistent by Hendrix as well. Thus she outlines that teaching methods should have maximum dynamic power for the learner in order to verbalize presented generalizations.

The article also concerns the pattern of the experiment and the most difficult aspects of its design. Besides, Hendrix presents one more problem faced in the course of carrying out the experiment which is evaluation of test results. However, the author managed to justify mentioned difficulties by means of repeated experiments. In addition, the author presents the table with the data helping the readers to understand clearly her experiment and research as whole.

It should be admitted that Gertrude Hendrix has an intention to proceed with the research since she presents interpretation and direction of the future inquiry. Thus she suggests discovery by deduction or learning generalization through arithmetic formulas. At the same time, there is a concern expressed by the author that teaching must not be authoritative. It is evident that Hendrix supports nondirective theory of instruction. However, she mentions that modern extremes lack implications of the theory of instruction.

All in all, the article of Gertrude Hendrix is quite critical one since it included a lot of denials and objections. Moreover, the hypotheses stated by Hendrix are to deny the assumption that verbalization is the representative example of general behavior. Along with such denial, the author explains that such critical examination of the problem is required and invited from anyone who is concerned.

## Discovery and expository teaching

The article “A Study of Discovery and Expository Presentation: Implications for Teaching” by Blaine R. Worthen is devoted to the discovery of learning, its advantages and disadvantages. The article’s strength is in presenting critics of discovery method in teaching, herewith supporting it by the claims of different scientists. Along with the discovery method, the author discusses expository method and comes to the conclusion that the most effective teaching is when using both expository and discovery teaching.

The research of Worthen is carried out on ground of a large experiment which is introduced by means of many tables in the article. Such amount of tables overload the article and make reader’s attention dispel. Moreover, he is weak at providing comprehensive explanations of the material presented in the tables.

Therefore, special attention should be paid to the measures of teaching behavior researched by means of two instruments. The author explains pupil perception of the teaching behavior with the help of the scoring method and brief directions. Moreover, Worthen makes teacher ratings in order to compare differences in teachers’ treatment. Thus the measures of teaching behavior are used as controls ensuring proper presentation of discovery and expository teaching.

In discussion and conclusions, results of teaching behavior are shown by the researcher. It should be outlined that his findings support the claims of many scientists concerning discovery methods. Thus article mostly concerns the drawbacks of the discovery methods and their influence on the pupils and their development.

In general, the analyzed article can be considered as a successful presentation of the research denying generally accepted theory of discovery method. Furthermore, Blaine R. Worthen managed to make his research interesting and appealing to the audience despite overloading his article with tables. All in all, the research introduced in the analyzed article is consistent and well-structured what makes readers perceive information faster and easier. Implications for future research are quite interesting and need initiations for exploring discovery-expository dimensions.

## Reference List

Bittinger M. (1968). A Review of Discovery. *The Mathematics Teacher*, pp. 140-146.

Gredler M. (2001). *Learning and Instruction*. *Theory into Practice (2*^{nd}* ed.).* New York: Macmillan Publishing Company.

Handa Y. (2003). A Phenomenological Exploration of Mathematical Engagement: Approaching an Old Metaphor anew. *For the Learning of Mathematics*, 23 (1): pp. 22-29.

Hendrix G.(1947). A New Clue to Transfer of Training. *The Elementary School Journal*, 48 (4): pp.197-208.

Worthen B. (1968). A Study of Discovery and Expository Presentation: Implications for Teaching. *Journal of Teacher Education*, 19: pp.223-242.