Dienes Theory in Mathematics Learning

This is the example of Dienes theory in mathematics learning about the number of diagonal in polygon. The purpose of the learning is to understand the number of diagonal in polygon.

Based on Dienes Theory, the stages are:
1.    Free Play

In this stage, students are given freedom by the teacher to draw triangle, quadrilateral, pentagon, and soon in their book.

2.   Games

In the stage, teacher gives a rule. The rules are:

• Draw diagonal of each polygon. As we know, a diagonal is a line segment connecting two nonadjacent vertices of a polygon.

• Find the number of diagonal in each polygon
  

3.    Searching For communalities

In this stage, students find the similarity about the number of each polygon.

4.   Representation

After students get the similarity about the number of each polygon, in this stage student can represent the figure as below.

5.   Symbolization

In this stage student symbolize the formula of pentagon’s diagonal number from the pattern that they get.

side	3	4	5	6	n	n
number of diagonal	3(3-3) = 0	4(4-3) = 0	5(5-3) = 0	6(6-3) = 0	……….	n(n-3) = 0

6.   Formulation

This is the last of the stage. Student can apply the general formula to another case, such as they can determine the number of polygon with side 23.

Guilford’s Theory

Guilford’s Theory

Intelligence Definition

Intelligence is an ability to act directionally, think rationally, and face the environment effectively. In generally it can be concluded that intelligence is a mental ability include thinking process rationally. Therefore, intelligence cannot perceive directly, but must be concluded from various real actions that represent by manifestation of rational thinking process.

Overview about Theories of Intelligence

1.Scientific Versus Lay Definitions
The meaning of intelligence is understood differently by psychologists and lay persons. Recent research shows that most lay persons think of intelligence as comprised of verbal ability, practical problem-solving ability, and social competence, example being fair with others, having a social conscience.
In contrast, experts define intelligence as including verbal ability, problem-solving ability, and practical intelligence but not the social competence that most laypersons apparently value.

2.Spearman's General factor
At the beginning of the 20th century British psychologist Charles Spearman theorized that every aptitude of people is caused by two factors, they are:
1.General factor(g)
2.Special factor(s)
General factor is factor as basis of the people aptitude. But special factor has function in special aptitude. Factor g has function in every people aptitude and factor s has function in different aptitude. He also give opinion that factor g appropriate on base and factor s appropriate on experiences(environment, education, etc. )

3.Thurstone's Primary Mental Abilities
The American psychologist L.L. Thurstone (1887 -1955) relied on the finding so fairly intelligence testing to develop his idea of primary mental abilities. According to Thurstone, there are seven such abilities necessary for high-level test performance, they are spatial ability; perceptual speed; numerical ability; verbal meaning; memory; word fluency; and reasoning.

4.Guilford's Three-Dimensional Model

About of J.P. Guilford
He was born on Nebraska, March 7, 1897, and died in Los Angeles on November 26, 1987. In 1924 Guilford entered the psychology Ph.D. program at Cornell University and was awarded in 1927. After short periods of time on the faculties of the universities of Illinois and Kansas, Guilford returned in 1928 to the University of Nebraska as professor of psychology, where he achieved an international reputation as one of America’s foremost psychologist. In 1940 he moved to the University of Southern California. Guilford was a psychologist involved during the World War II in developing tests to select candidates for training as pilots. As he expanded his interests into testing various other specific thinking skills, he developed a model to guide his research and to organize his thinking about all the various skills he was testing.

J.P. Guilford developed the idea of specific intelligence factors into a very detailed model beginning in the 1950s. Guilford(1967) conceives of intelligence as being a combination of three dimensions, they are:
1. Operations
2. Contents
3. Products

With each specific intellectual ability being one specific combination of these three factors. Since there are 5 operations, 5 contents, and 6 products in Guilford’s model. This means that there are 150 different mental abilities that could conceivably be measured. Although only about 100 have been measured to date, Guilford’s model is an important advance in understanding intelligence.

Operations
a.Cognition is the ability to recognize various forms of information and to understand information.
Example: A child who can separate a mixed pile of squares and triangles into separate piles of squares and triangles is exercising a degree of cognition.
b.Memory is the ability to store information in the mind and to call out stored information in response to certain stimuli. Two kinds of memory:
Memory retention is contain memory related to daily life
Memory recording is fresh memories
Example: A student who immediately answer 1 when asked to give the sine of 90° is using his or her memory.
c.Divergent is the ability to view given information in a new way so that unique and unexpected conclusions are the consequence.
Example: A mathematician who discovers and proves a new and important mathematical theorem is exhibiting considerable ability in divergent production.
d.Convergent is the ability to take a specified set of information and draw a universally accepted conclusion or response based upon the given information.
Example: In algebra lesson, student who finds the correct solution to a set of three linear equations have used his or her convergent production ability.
e.Evaluation is the ability to process information in order to make judgments, draw conclusion and arrive at decisions.
Example: If we want to solve mathematics problem we think hard to solve it by simple method

Contents
a.Visual is information in visual form such as are shape or color.
example: triangle, cubes, parabola, etc.
b.Auditory involves information in auditory form, such as spoken words or music
c.Symbolic are symbol or codes representing concrete object or abstract concepts.
Example: + is the mathematical symbol for the operation of addition.
d.Semantic of learning are those words and ideas which evoke a mental image when they are presented as stimuli.
Example: sun, car, white, moon, etc. are word which evoke image in people’s minds when they hear or read them.
e.Behavioral contents of learning are the manifestations of stimuli and responses in people can be also obtained through facial expression or voice.

Products
a.Unit is a single symbol, figure, word, object, or idea.
Example: each real number.
b.Classes is sets of items grouped by virtue of their common properties
Example: set of real numbers.
c.Relations are connections between items of information
Example: equality and inequality are relation in the set of real numbers.
d.A systems is a composition of units, classes, and relationship into a larger and more meaningful structure.
Example: the set of real numbers together with the operations of addition, subs traction, multiplication, and division and the algebraic properties of these operations.
e.Transformation is the process of modifying, reinterpreting, and restructuring existing information into new information. The transformation ability is usually thought to be characteristic of creative people.
Example: functions defined on the real number system.
f.An implication is a prediction or a conjecture about the consequences of interactions among units, classes, relations, systems, and transformations.
Example: each theorem about function on the real numbers.

Application of Guilford Theory in Math Education

Example task of creativity that was developed by Guilford in applied starting at the kindergarten level, they are recognizing numbers, drawing plane and space. At the elementary and secondary school level even in the university level, there are some essential materials allows children to make them creative such as the geometry material.

Examples of material to determine the creativity of student

Factors that Influence Intelligence

References
1. Robert R Reilly/Ernest L. 1981. Lewis. Educational Psychology. New York : Macmillan Publishing Co
2. Arno F. Witting and Gurney Williams III. 1984. Psychology an Introduction. Mc Graw Hill Book Co
3. www.drchrustowski.com/Intelligence.pdf

Assessing Student Attitudes

Importance of student Attitude

All learning has affective components. Whenever a student masters knowledge or skills, she or he develops an attitude toward subject area and the processes of learning. Because student’s attitudes influence future behavior, the development of positive attitudes may be more important than mastery of specific knowledge and skills.

The overall purpose of schools is to develop each student to maximum capacity as a productive and happy member of society. An important measure of success is not the degree to which student’s master knowledge and skills, but whether the students voluntarily use such knowledge and skills in their daily life outside of school and in their lives after they have finished school.  

Besides positive attitudes toward subject areas and skills such as reading, writing, and math, schools are supposed to inculcate positive attitude toward:

1)      self

2)      diverse Others

3)      potential careers

4)      being role responsible and role readiness 

The most important of all, schools are supposed to ensure students develop positive attitudes toward our pluralistic, democratic society, freedom of choice, equality of opportunity, self-reliance, and free and open inquiry into all issues

An attitude is a positive or negative reaction to a person, object, or idea. It is a learned predisposition to respond in favorable or unfavorable manner to a particular person, object or idea. Attitude is important determinants of behavior.

A teacher, school, or school system can asses student attitudes through observational procedures, interviews, and questionnaires.  

In planning how to use questionnaires to measure student attitude, you may use the following procedure:

•          Decide on which attitudes to measure

•   Construct a questionnaires by writing specific questions to measure the targeted attitudes

•          Select the standardized attitude measures you want to use, if any

•    Give your questionnaire near the beginning and then near the end of an instructional unit, semester, or year

•   Analyze and organize the data for feedback to interested stakeholders to make instructional decision

•      Give the feedback in a timely and orderly way and facilitate stakeholders’ use of the data

•    Use the data on student attitudes to modify and improve the course and your teaching

Deciding Which Attitudes to Measure

Which attitudes you want to measure depends on your instructional goals and the subject you teach. Minimally, however, you may want to measure attitudes toward the subject area, the instructional method used, and learning in general.

Constructing Your Own Questionnaire

In preparing your own questionnaire, there are three types of questions can be used, they are:

1.       Open-Ended Question

2.       Closed-Ended Question

3.       Semantic Differential Question

Open-Ended Question

Open-Ended Questions call for the student to answer by writing a statement that may vary in length. They may require respondents to give a free response or supply a word or phrase to fill in the blank.

Example of open-ended questions follows:

•          My General opinion about Mathematics is …………………………….

•          My teachers are …………………………

•          If someone suggested I take up American history as my life’s work, I would reply, ……………………..

•          Mathematics is my …………………………………….. subject.

Open-ended questions are a good way to obtain new ideas about what to ask to measure student attitude and values.  Student responses are scored by counting the number of times a word or phrase occurs. A mean and standard deviation may then be calculated. Open-ended questions, however, tend to be hard to analyze and often are not fully answered.

Closed-Ended Question

Closed-ended questions require the student to indicate the alternative answer closest to his or her internal response. The response they require can be dichotomous, multiple choice, ranking, or scale. Here are some examples.

·         Do you intend to take another course in Mathematics ……… Yes …….. No

·         Circle each of the words that tell how you feel about Mathematics:

        Interesting                      very important               worthless           

        Boring                            difficult                        useful  

·         Rank these subject areas from most interesting (1) to least interesting (6) to you :

        Social Studies                                     English

        Science                                                 Mathematics

        Physical Education                           Foreign Language

·         How interested are you in learning more about Mathematics?

        Very uninterested       1   2   3   4   5   6   7       Very Interested

The questions are scored by counting the frequencies of each response and then calculating the mean response and the standard deviation

Semantic Differential Question

This type of question allows the teacher to present any object (be it person, issue, practice, subject area, or anything else) and obtain an indication of student attitude toward it. It’s consist of a series of rating scales of bipolar adjectives pairs describing a concept the teacher wants to obtain student attitudes toward. An example is:

POETRY

Ugly                          1   2   3   4   5   6   7           Beautiful

Bad                          1   2   3   4   5   6   7           Good

Worthless                  1   2   3   4   5   6   7           Valuable

Negative                   1   2   3   4   5   6   7           Positive

How good is your question?

Writing good open-ended and closed-ended questions takes some expertise and practice. To evaluate each question you should considering the following points:

1)      Is the question worded simply with no abbreviations and difficult word?

2)      Are all the words in the question familiar to the respondents?

3)      Is the question worded without slang spaces, colloquialisms, and bureaucratic words?

4)      Are any words emotionally loaded, vaguely defined, or overly general?

5)      Does the question have unstated assumptions or implications that lead respondents to give certain response?

6)      Does the question presuppose a certain state affairs?

7)      Does the question ask for only one bit of information?

8)      Is only one adjective or adverb used in the question?

9)      Does the wording of the question imply a desired answer?

10)   Do any words have a double meaning that may cause misunderstanding?

11)   Are the response options mutually exclusive and sufficient  to cover each conceivable answer?

12)   Does the question contain words that tend not to have a common meaning, such as significant, always, usually, most, never, and several?

Deciding on types of responses to questions?

There are two types of responses, they are:

ü  Open-ended responses

The types of open-ended responses fill in the blank and free responses.

ü  Close-ended

The types of closed-ended responses include dichotomous, multiple choice, ranking, or scale.

Deciding on types of responses to questions?

The following list of questions will help you to determine which type of response you want to elicit

1.       Is the question best asked as an open-ended or closed ended question?

2.       For open-ended questions, should respondents give a free response or fill in the blank?

3.       For closed-ended questions, should the response be dichotomous, multiple, choice, ranking, scale?

4.       If a multiple-choice or scale check response is used, does it (a)cover adequately all the significant alternatives without overlapping,(b)provide choices in a defensible order, and (c) provide response alternatives of uniform value(distance)

One of the most common approaches for writing closed-ended question with a scaled response is the Likert method of summated rating. The procedure for developing a Likert scale is to ask several questions about the topic of interest. For each question a response scale is given with anywhere from three to nine points. Questions with five alternatives are quite common. Two ways of presenting the alternatives follow.

Strongly disagree (1)                      Agree (4)

Disagree (2)                                        strongly agree (5)

Undecided (3)

Strongly disagree                1   2   3   4   5     strongly Agree

A student’s responses for all the questions are summed together to get an overall scale indicating the student’s attitudes toward the issue being measured.

To assess the quality of your questions against the following criteria. 

1.       Is the question necessary? The question may ask for information that is (a) already covered in another question or(b) more detailed that necessary

2.       Does the question cover too much material?

3.       Do respondents have the information necessary to answer the question?

4.       Does the question need to be more concrete, specific, and closely related to the respondent’s personal experience?

5.       Is the question content sufficiently general and free from spurious concreteness and specify?

6.       Is the question content biased or loaded in one direction without accompanying questions to balance the emphasis?

7.       Will the respondents give the information sought?